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The Wielandt subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201701.png" /> of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201702.png" /> is defined to be the intersection of the normalizers of all the subnormal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201703.png" /> (cf. also [[Subnormal subgroup|Subnormal subgroup]]; [[Normal subgroup|Normal subgroup]]; [[Normalizer of a subset|Normalizer of a subset]]). This characteristic subgroup was introduced in 1958 by H. Wielandt [[#References|[a18]]]. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201704.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201705.png" /> is a T-group, i.e. normality is transitive in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201706.png" />.
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When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201707.png" /> is nilpotent (cf. [[Nilpotent group|Nilpotent group]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201708.png" /> is the intersection of the normalizers of all the subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w1201709.png" />. The latter is called the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017010.png" /> and was introduced by R. Baer [[#References|[a1]]]. In [[#References|[a17]]], E. Schenkman showed that the norm is always contained in the second centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017011.png" /> (cf. also [[Centre of a group|Centre of a group]]). Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017012.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017013.png" /> is a nilpotent group.
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The upper Wielandt series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017014.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017015.png" /> is formed by iteration; thus,
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The Wielandt subgroup $\omega ( G )$ of a [[Group|group]] $G$ is defined to be the intersection of the normalizers of all the subnormal subgroups of $G$ (cf. also [[Subnormal subgroup|Subnormal subgroup]]; [[Normal subgroup|Normal subgroup]]; [[Normalizer of a subset|Normalizer of a subset]]). This characteristic subgroup was introduced in 1958 by H. Wielandt [[#References|[a18]]]. Note that $\omega ( G ) = G$ if and only if $G$ is a [[T-group]], i.e. normality is transitive in $G$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017016.png" /></td> </tr></table>
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When $G$ is nilpotent (cf. [[Nilpotent group|Nilpotent group]]), $\omega ( G )$ is the intersection of the normalizers of all the subgroups of $G$. The latter is called the norm of $G$ and was introduced by R. Baer [[#References|[a1]]]. In [[#References|[a17]]], E. Schenkman showed that the norm is always contained in the second centre $Z _ { 2 } ( G )$ (cf. also [[Centre of a group|Centre of a group]]). Hence $Z ( G ) \leq \omega ( G ) \leq Z _ { 2 } ( G )$ if $G$ is a nilpotent group.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017017.png" /></td> </tr></table>
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The upper Wielandt series $\{ \omega _ { \alpha } ( G ) \}$ of a group $G$ is formed by iteration; thus,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017018.png" /> is an ordinal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017019.png" /> a limit ordinal (cf. also [[Ordinal number|Ordinal number]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017020.png" /> for some ordinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017021.png" />, the smallest such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017022.png" /> is called the Wielandt length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017024.png" /> has finite Wielandt length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017025.png" />, then every subnormal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017026.png" /> has defect at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017027.png" />.
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\begin{equation*} \omega _ { 0 } ( G ) = 1 \end{equation*}
  
Wielandt [[#References|[a18]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017028.png" /> contains all the subnormal (non-Abelian) simple subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017029.png" />, and also all the minimal normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017030.png" /> that satisfy min-sn, the minimal condition on subnormal subgroups (cf. also [[Group with the minimum condition|Group with the minimum condition]]). This last result implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017031.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017032.png" /> is a non-trivial group with min-sn. Thus, the Wielandt subgroup is unexpectedly large for groups with min-sn.
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\begin{equation*} \omega _ { \alpha + 1 } ( G ) / \omega _ { \alpha } ( G ) = \omega ( G / \omega _ { \alpha } ( G ) ) , \, \omega _ { \lambda } ( G ) = \cup _ { \beta &lt; \lambda } \omega _ { \beta } ( G ), \end{equation*}
  
A stronger result in this direction was found independently by D.J.S. Robinson [[#References|[a15]]] and J.E. Roseblade [[#References|[a16]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017033.png" /> is a group with min-sn, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017034.png" /> is finite. Thus, a subnormal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017035.png" /> has only finitely many conjugates. In addition, O.H. Kegel [[#References|[a11]]] generalized the first of Wielandt's results by demonstrating that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017036.png" /> contains all subnormal perfect T-subgroups of of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017037.png" />. Since solvable T-groups are metAbelian ([[#References|[a14]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017038.png" /> is a perfect T-group. It follows that the join (cf. also [[Join|Join]]) of all the subnormal perfect T-subgroups coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017039.png" /> in any group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017040.png" />. For a smooth treatment of these results see [[#References|[a12]]].
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where $\alpha$ is an ordinal and $\lambda$ a limit ordinal (cf. also [[Ordinal number|Ordinal number]]). If $G = \omega _ { \alpha } ( G )$ for some ordinal $\alpha$, the smallest such $\alpha$ is called the Wielandt length of $G$. If $G$ has finite Wielandt length $n$, then every subnormal subgroup of $G$ has defect at most $n$.
  
The example of the infinite dihedral group shows that the Wielandt subgroup of a polycyclic group can easily be trivial. In 1991, J. Cossey [[#References|[a7]]] showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017041.png" /> is a [[Polycyclic group|polycyclic group]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017042.png" /> is finite. Cossey also proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017043.png" /> is finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017044.png" /> is nilpotent-by-Abelian. Subsequently, this conclusion was extended to polycyclic groups that are meta-nilpotent or Abelian-by-finite by R. Brandl, S. Franciosi and F. de Giovanni. These authors were also able to show that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017045.png" /> is a finitely generated solvable-by-finite group of finite Prüfer rank (cf. [[Rank|Rank]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017046.png" /> is contained in the FC-centre (cf. also [[Group with a finiteness condition|Group with a finiteness condition]]).
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Wielandt [[#References|[a18]]] proved that $\omega ( G )$ contains all the subnormal (non-Abelian) simple subgroups of $G$, and also all the minimal normal subgroups of $G$ that satisfy min-sn, the minimal condition on subnormal subgroups (cf. also [[Group with the minimum condition|Group with the minimum condition]]). This last result implies that $\omega ( G ) \neq 1$ whenever $G$ is a non-trivial group with min-sn. Thus, the Wielandt subgroup is unexpectedly large for groups with min-sn.
  
In some ways <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017047.png" /> might seem to be close to the centre for polycyclic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017048.png" />. However, Cossey [[#References|[a7]]] has pointed out that there is a nilpotent-by-finite polycyclic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017050.png" /> is non-trivial and free Abelian. Cossey [[#References|[a7]]] has also investigated the Wielandt length of a polycyclic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017051.png" />, showing that this exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017052.png" /> is finite-by-nilpotent (when, of course, the Wielandt length is finite).
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A stronger result in this direction was found independently by D.J.S. Robinson [[#References|[a15]]] and J.E. Roseblade [[#References|[a16]]]: If $G$ is a group with min-sn, then $G / \omega ( G )$ is finite. Thus, a subnormal subgroup of $G$ has only finitely many conjugates. In addition, O.H. Kegel [[#References|[a11]]] generalized the first of Wielandt's results by demonstrating that $\omega ( G )$ contains all subnormal perfect T-subgroups of $G$. Since solvable T-groups are [[Meta-Abelian group|metAbelian]] ([[#References|[a14]]]), $\omega ^ { \prime \prime } ( G )$ is a perfect T-group. It follows that the join (cf. also [[Join|Join]]) of all the subnormal perfect T-subgroups coincides with $\omega ^ { \prime \prime } ( G )$ in any group $G$. For a smooth treatment of these results see [[#References|[a12]]].
  
The Wielandt length of finite solvable groups has been investigated extensively by several authors, and its connection with other invariants such as the derived length and nilpotent length has been analyzed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017053.png" /> be a finite [[Solvable group|solvable group]] with Wielandt length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017054.png" />. Since solvable T-groups are metAbelian, the derived length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017056.png" /> is at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017057.png" />. A.R. Camina [[#References|[a5]]] improved this bound by showing that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017059.png" /> is the nilpotent length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017061.png" />. Camina also showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017062.png" />. Subsequently, R. Bryce and J. Cossey [[#References|[a4]]] gave the improved estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017063.png" />, which is best possible when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017064.png" />. This work has been extended by C. Casolo [[#References|[a6]]] to infinite solvable groups with finite Wielandt length. For more results on the Wielandt length of finite groups, see [[#References|[a8]]].
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The example of the infinite dihedral group shows that the Wielandt subgroup of a polycyclic group can easily be trivial. In 1991, J. Cossey [[#References|[a7]]] showed that if $G$ is a [[Polycyclic group|polycyclic group]], then $G / C _ { G } ( \omega ( G ) )$ is finite. Cossey also proved that $\omega ( G ) / Z ( G )$ is finite if $G$ is nilpotent-by-Abelian. Subsequently, this conclusion was extended to polycyclic groups that are [[Meta-nilpotent group|meta-nilpotent]] or Abelian-by-finite by R. Brandl, S. Franciosi and F. de Giovanni. These authors were also able to show that if $G$ is a finitely generated solvable-by-finite group of finite Prüfer rank (cf. [[Rank|Rank]]), then $\omega ( G )$ is contained in the FC-centre (cf. also [[Group with a finiteness condition|Group with a finiteness condition]]).
  
In [[#References|[a4]]], a local version of the Wielandt subgroup was introduced, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017065.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017066.png" /> is a prime number; this is the intersection of the normalizers of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017067.png" />-perfect subnormal subgroups, i.e. those that have no non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017068.png" />-quotients. They showed that
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In some ways $\omega ( G )$ might seem to be close to the centre for polycyclic groups $G$. However, Cossey [[#References|[a7]]] has pointed out that there is a nilpotent-by-finite polycyclic group $G$ such that $\omega ( G ) / Z ( G )$ is non-trivial and free Abelian. Cossey [[#References|[a7]]] has also investigated the Wielandt length of a polycyclic group $G$, showing that this exists if and only if $G$ is finite-by-nilpotent (when, of course, the Wielandt length is finite).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017069.png" /></td> </tr></table>
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The Wielandt length of finite solvable groups has been investigated extensively by several authors, and its connection with other invariants such as the derived length and nilpotent length has been analyzed. Let $G$ be a finite [[Solvable group|solvable group]] with Wielandt length $l$. Since solvable T-groups are metAbelian, the derived length $d$ of $G$ is at most $2 l$. A.R. Camina [[#References|[a5]]] improved this bound by showing that $d \leq l + n - 1$, where $n$ is the nilpotent length of $G$ and $l + n &gt; 2$. Camina also showed that $n \leq l + 1$. Subsequently, R. Bryce and J. Cossey [[#References|[a4]]] gave the improved estimate $d \leq ( 5 l + 2 ) / 3$, which is best possible when $l \equiv 2 ( \operatorname { mod } 3 )$. This work has been extended by C. Casolo [[#References|[a6]]] to infinite solvable groups with finite Wielandt length. For more results on the Wielandt length of finite groups, see [[#References|[a8]]].
  
Another variation of the Wielandt subgroup has been described in [[#References|[a2]]]. The generalized Wielandt subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017070.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017071.png" /> is defined to be the intersection of the normalizers of all the infinite subnormal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017072.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017073.png" /> precisely when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017074.png" /> is an IT-group, i.e. all infinite subnormal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017075.png" /> are normal. The class of IT-groups has been investigated in [[#References|[a10]]] and [[#References|[a9]]]. J.C. Beidleman, M.R. Dixon and D.J.S. Robinson [[#References|[a2]]] showed that the subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017077.png" /> are remarkably close, and indeed they coincide in many cases. In [[#References|[a2]]] the following results were established. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017078.png" /> be an infinite group; then:
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In [[#References|[a4]]], a local version of the Wielandt subgroup was introduced, $\omega ^ { p } ( G )$ where $p$ is a prime number; this is the intersection of the normalizers of the $p ^ { \prime }$-perfect subnormal subgroups, i.e. those that have no non-trivial $p ^ { \prime }$-quotients. They showed that
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017079.png" /> is a residually finite T-group (cf. also [[Residually-finite group|Residually-finite group]]);
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\begin{equation*} \omega ( G ) = \bigcap _ { p } \omega ^ { p } ( G ). \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017080.png" /> unless the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017081.png" /> generated by all the finite solvable subnormal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017082.png" /> is Prüfer-by-finite;
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Another variation of the Wielandt subgroup has been described in [[#References|[a2]]]. The generalized Wielandt subgroup $\iota \ \omega ( G )$ of a group $G$ is defined to be the intersection of the normalizers of all the infinite subnormal subgroups of $G$. Thus, $\iota \omega ( G ) = G$ precisely when $G$ is an IT-group, i.e. all infinite subnormal subgroups of $G$ are normal. The class of IT-groups has been investigated in [[#References|[a10]]] and [[#References|[a9]]]. J.C. Beidleman, M.R. Dixon and D.J.S. Robinson [[#References|[a2]]] showed that the subgroups $\omega ( G )$ and $\iota \ \omega ( G )$ are remarkably close, and indeed they coincide in many cases. In [[#References|[a2]]] the following results were established. Let $G$ be an infinite group; then:
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017083.png" /> is Prüfer-by-finite and infinite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017084.png" /> is metAbelian;
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$\iota \ \omega ( G ) / \omega ( G )$ is a residually finite T-group (cf. also [[Residually-finite group|Residually-finite group]]);
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017085.png" /> is finite, so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017086.png" />. Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017087.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017088.png" /> is a finitely generated infinite solvable group.
+
$\iota \omega ( G ) = \omega ( G )$ unless the subgroup $S$ generated by all the finite solvable subnormal subgroups of $G$ is Prüfer-by-finite;
 +
 
 +
if $S$ is Prüfer-by-finite and infinite, then $\iota \ \omega ( G ) / \omega ( G )$ is metAbelian;
 +
 
 +
if $S$ is finite, so is $\iota \ \omega ( G ) / \omega ( G )$. Finally, $\iota \omega ( G ) = \omega ( G )$ if $G$ is a finitely generated infinite solvable group.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Baer,  "Der Kern, eine charakteristische Untergruppe"  ''Compositio Math.'' , '''1'''  (1934)  pp. 254–283</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. Beidleman,  M.R. Dixon,  D.J.S. Robinson,  "The generalized Wielandt subgroup of a group"  ''Canad. J. Math.'' , '''47''' :  2  (1995)  pp. 246–261</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Brandl,  S. Franciosi,  F. de Giovanni,  "On the Wielandt subgroup of infinite soluble groups"  ''Glasgow Math. J.'' , '''32'''  (1990)  pp. 121–125</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.A. Bryce,  J. Cossey,  "The Wielandt subgroup of a finite soluble group"  ''J. London Math. Soc. (2)'' , '''40'''  (1989)  pp. 244–256</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.R. Camina,  "The Wielandt length of finite groups"  ''J. Algebra'' , '''15'''  (1970)  pp. 142–148</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Casolo,  "Soluble groups with finite Wielandt length"  ''Glasgow Math. J.'' , '''31'''  (1989)  pp. 329–334</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Cossey,  "The Wielandt subgroup of a polycyclic group"  ''Glasgow Math. J.'' , '''33'''  (1991)  pp. 231–234</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Cossey,  "Finite groups generated by subnormal T-subgroups"  ''Glasgow Math. J.'' , '''37'''  (1995)  pp. 363–371</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  F. de Giovanni,  S. Franciosi,  "Groups in which every infinite subnormal subgroup is normal"  ''J. Algebra'' , '''96'''  (1985)  pp. 566–580</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H. Heineken,  "Groups with restrictions on their infinite subnormal subgroups"  ''Proc. Edinburgh Math. Soc.'' , '''31'''  (1988)  pp. 231–241</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  O.H. Kegel,  "Über den Normalisator von subnormalen und erreichbaren Untergruppen"  ''Math. Ann.'' , '''163'''  (1966)  pp. 248–258</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J.C. Lennox,  S.E. Stonehewer,  "Subnormal subgroups of groups" , Oxford  (1987)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  E. Ormerod,  "The Wielandt subgroup of a metacyclic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120170/w12017089.png" />-group"  ''Bull. Austral. Math. Soc.'' , '''42'''  (1990)  pp. 499–510</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  D.J.S. Robinson,  "Groups in which normality is a transitive relation"  ''Proc. Cambridge Philos. Soc.'' , '''60'''  (1964)  pp. 21–38</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  D.J.S. Robinson,  "On the theory of subnormal subgroups"  ''Math. Z.'' , '''89'''  (1965)  pp. 30–51</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  J.E. Roseblade,  "On certain subnormal coalition classes"  ''J. Algebra'' , '''1'''  (1964)  pp. 132–138</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  E. Schenkman,  "On the norm of a group"  ''Illinois J. Math.'' , '''4'''  (1960)  pp. 150–152</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  H. Wielandt,  "Über den Normalisator der subnormalen Untergruppen"  ''Math. Z.'' , '''69'''  (1958)  pp. 463–465</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  R. Baer,  "Der Kern, eine charakteristische Untergruppe"  ''Compositio Math.'' , '''1'''  (1934)  pp. 254–283</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.C. Beidleman,  M.R. Dixon,  D.J.S. Robinson,  "The generalized Wielandt subgroup of a group"  ''Canad. J. Math.'' , '''47''' :  2  (1995)  pp. 246–261</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  R. Brandl,  S. Franciosi,  F. de Giovanni,  "On the Wielandt subgroup of infinite soluble groups"  ''Glasgow Math. J.'' , '''32'''  (1990)  pp. 121–125</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R.A. Bryce,  J. Cossey,  "The Wielandt subgroup of a finite soluble group"  ''J. London Math. Soc. (2)'' , '''40'''  (1989)  pp. 244–256</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A.R. Camina,  "The Wielandt length of finite groups"  ''J. Algebra'' , '''15'''  (1970)  pp. 142–148</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C. Casolo,  "Soluble groups with finite Wielandt length"  ''Glasgow Math. J.'' , '''31'''  (1989)  pp. 329–334</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J. Cossey,  "The Wielandt subgroup of a polycyclic group"  ''Glasgow Math. J.'' , '''33'''  (1991)  pp. 231–234</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J. Cossey,  "Finite groups generated by subnormal T-subgroups"  ''Glasgow Math. J.'' , '''37'''  (1995)  pp. 363–371</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  F. de Giovanni,  S. Franciosi,  "Groups in which every infinite subnormal subgroup is normal"  ''J. Algebra'' , '''96'''  (1985)  pp. 566–580</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  H. Heineken,  "Groups with restrictions on their infinite subnormal subgroups"  ''Proc. Edinburgh Math. Soc.'' , '''31'''  (1988)  pp. 231–241</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  O.H. Kegel,  "Über den Normalisator von subnormalen und erreichbaren Untergruppen"  ''Math. Ann.'' , '''163'''  (1966)  pp. 248–258</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J.C. Lennox,  S.E. Stonehewer,  "Subnormal subgroups of groups" , Oxford  (1987)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  E. Ormerod,  "The Wielandt subgroup of a metacyclic $p$-group"  ''Bull. Austral. Math. Soc.'' , '''42'''  (1990)  pp. 499–510</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  D.J.S. Robinson,  "Groups in which normality is a transitive relation"  ''Proc. Cambridge Philos. Soc.'' , '''60'''  (1964)  pp. 21–38</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  D.J.S. Robinson,  "On the theory of subnormal subgroups"  ''Math. Z.'' , '''89'''  (1965)  pp. 30–51</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  J.E. Roseblade,  "On certain subnormal coalition classes"  ''J. Algebra'' , '''1'''  (1964)  pp. 132–138</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  E. Schenkman,  "On the norm of a group"  ''Illinois J. Math.'' , '''4'''  (1960)  pp. 150–152</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  H. Wielandt,  "Über den Normalisator der subnormalen Untergruppen"  ''Math. Z.'' , '''69'''  (1958)  pp. 463–465</td></tr></table>

Latest revision as of 09:23, 21 January 2021

The Wielandt subgroup $\omega ( G )$ of a group $G$ is defined to be the intersection of the normalizers of all the subnormal subgroups of $G$ (cf. also Subnormal subgroup; Normal subgroup; Normalizer of a subset). This characteristic subgroup was introduced in 1958 by H. Wielandt [a18]. Note that $\omega ( G ) = G$ if and only if $G$ is a T-group, i.e. normality is transitive in $G$.

When $G$ is nilpotent (cf. Nilpotent group), $\omega ( G )$ is the intersection of the normalizers of all the subgroups of $G$. The latter is called the norm of $G$ and was introduced by R. Baer [a1]. In [a17], E. Schenkman showed that the norm is always contained in the second centre $Z _ { 2 } ( G )$ (cf. also Centre of a group). Hence $Z ( G ) \leq \omega ( G ) \leq Z _ { 2 } ( G )$ if $G$ is a nilpotent group.

The upper Wielandt series $\{ \omega _ { \alpha } ( G ) \}$ of a group $G$ is formed by iteration; thus,

\begin{equation*} \omega _ { 0 } ( G ) = 1 \end{equation*}

\begin{equation*} \omega _ { \alpha + 1 } ( G ) / \omega _ { \alpha } ( G ) = \omega ( G / \omega _ { \alpha } ( G ) ) , \, \omega _ { \lambda } ( G ) = \cup _ { \beta < \lambda } \omega _ { \beta } ( G ), \end{equation*}

where $\alpha$ is an ordinal and $\lambda$ a limit ordinal (cf. also Ordinal number). If $G = \omega _ { \alpha } ( G )$ for some ordinal $\alpha$, the smallest such $\alpha$ is called the Wielandt length of $G$. If $G$ has finite Wielandt length $n$, then every subnormal subgroup of $G$ has defect at most $n$.

Wielandt [a18] proved that $\omega ( G )$ contains all the subnormal (non-Abelian) simple subgroups of $G$, and also all the minimal normal subgroups of $G$ that satisfy min-sn, the minimal condition on subnormal subgroups (cf. also Group with the minimum condition). This last result implies that $\omega ( G ) \neq 1$ whenever $G$ is a non-trivial group with min-sn. Thus, the Wielandt subgroup is unexpectedly large for groups with min-sn.

A stronger result in this direction was found independently by D.J.S. Robinson [a15] and J.E. Roseblade [a16]: If $G$ is a group with min-sn, then $G / \omega ( G )$ is finite. Thus, a subnormal subgroup of $G$ has only finitely many conjugates. In addition, O.H. Kegel [a11] generalized the first of Wielandt's results by demonstrating that $\omega ( G )$ contains all subnormal perfect T-subgroups of $G$. Since solvable T-groups are metAbelian ([a14]), $\omega ^ { \prime \prime } ( G )$ is a perfect T-group. It follows that the join (cf. also Join) of all the subnormal perfect T-subgroups coincides with $\omega ^ { \prime \prime } ( G )$ in any group $G$. For a smooth treatment of these results see [a12].

The example of the infinite dihedral group shows that the Wielandt subgroup of a polycyclic group can easily be trivial. In 1991, J. Cossey [a7] showed that if $G$ is a polycyclic group, then $G / C _ { G } ( \omega ( G ) )$ is finite. Cossey also proved that $\omega ( G ) / Z ( G )$ is finite if $G$ is nilpotent-by-Abelian. Subsequently, this conclusion was extended to polycyclic groups that are meta-nilpotent or Abelian-by-finite by R. Brandl, S. Franciosi and F. de Giovanni. These authors were also able to show that if $G$ is a finitely generated solvable-by-finite group of finite Prüfer rank (cf. Rank), then $\omega ( G )$ is contained in the FC-centre (cf. also Group with a finiteness condition).

In some ways $\omega ( G )$ might seem to be close to the centre for polycyclic groups $G$. However, Cossey [a7] has pointed out that there is a nilpotent-by-finite polycyclic group $G$ such that $\omega ( G ) / Z ( G )$ is non-trivial and free Abelian. Cossey [a7] has also investigated the Wielandt length of a polycyclic group $G$, showing that this exists if and only if $G$ is finite-by-nilpotent (when, of course, the Wielandt length is finite).

The Wielandt length of finite solvable groups has been investigated extensively by several authors, and its connection with other invariants such as the derived length and nilpotent length has been analyzed. Let $G$ be a finite solvable group with Wielandt length $l$. Since solvable T-groups are metAbelian, the derived length $d$ of $G$ is at most $2 l$. A.R. Camina [a5] improved this bound by showing that $d \leq l + n - 1$, where $n$ is the nilpotent length of $G$ and $l + n > 2$. Camina also showed that $n \leq l + 1$. Subsequently, R. Bryce and J. Cossey [a4] gave the improved estimate $d \leq ( 5 l + 2 ) / 3$, which is best possible when $l \equiv 2 ( \operatorname { mod } 3 )$. This work has been extended by C. Casolo [a6] to infinite solvable groups with finite Wielandt length. For more results on the Wielandt length of finite groups, see [a8].

In [a4], a local version of the Wielandt subgroup was introduced, $\omega ^ { p } ( G )$ where $p$ is a prime number; this is the intersection of the normalizers of the $p ^ { \prime }$-perfect subnormal subgroups, i.e. those that have no non-trivial $p ^ { \prime }$-quotients. They showed that

\begin{equation*} \omega ( G ) = \bigcap _ { p } \omega ^ { p } ( G ). \end{equation*}

Another variation of the Wielandt subgroup has been described in [a2]. The generalized Wielandt subgroup $\iota \ \omega ( G )$ of a group $G$ is defined to be the intersection of the normalizers of all the infinite subnormal subgroups of $G$. Thus, $\iota \omega ( G ) = G$ precisely when $G$ is an IT-group, i.e. all infinite subnormal subgroups of $G$ are normal. The class of IT-groups has been investigated in [a10] and [a9]. J.C. Beidleman, M.R. Dixon and D.J.S. Robinson [a2] showed that the subgroups $\omega ( G )$ and $\iota \ \omega ( G )$ are remarkably close, and indeed they coincide in many cases. In [a2] the following results were established. Let $G$ be an infinite group; then:

$\iota \ \omega ( G ) / \omega ( G )$ is a residually finite T-group (cf. also Residually-finite group);

$\iota \omega ( G ) = \omega ( G )$ unless the subgroup $S$ generated by all the finite solvable subnormal subgroups of $G$ is Prüfer-by-finite;

if $S$ is Prüfer-by-finite and infinite, then $\iota \ \omega ( G ) / \omega ( G )$ is metAbelian;

if $S$ is finite, so is $\iota \ \omega ( G ) / \omega ( G )$. Finally, $\iota \omega ( G ) = \omega ( G )$ if $G$ is a finitely generated infinite solvable group.

References

[a1] R. Baer, "Der Kern, eine charakteristische Untergruppe" Compositio Math. , 1 (1934) pp. 254–283
[a2] J.C. Beidleman, M.R. Dixon, D.J.S. Robinson, "The generalized Wielandt subgroup of a group" Canad. J. Math. , 47 : 2 (1995) pp. 246–261
[a3] R. Brandl, S. Franciosi, F. de Giovanni, "On the Wielandt subgroup of infinite soluble groups" Glasgow Math. J. , 32 (1990) pp. 121–125
[a4] R.A. Bryce, J. Cossey, "The Wielandt subgroup of a finite soluble group" J. London Math. Soc. (2) , 40 (1989) pp. 244–256
[a5] A.R. Camina, "The Wielandt length of finite groups" J. Algebra , 15 (1970) pp. 142–148
[a6] C. Casolo, "Soluble groups with finite Wielandt length" Glasgow Math. J. , 31 (1989) pp. 329–334
[a7] J. Cossey, "The Wielandt subgroup of a polycyclic group" Glasgow Math. J. , 33 (1991) pp. 231–234
[a8] J. Cossey, "Finite groups generated by subnormal T-subgroups" Glasgow Math. J. , 37 (1995) pp. 363–371
[a9] F. de Giovanni, S. Franciosi, "Groups in which every infinite subnormal subgroup is normal" J. Algebra , 96 (1985) pp. 566–580
[a10] H. Heineken, "Groups with restrictions on their infinite subnormal subgroups" Proc. Edinburgh Math. Soc. , 31 (1988) pp. 231–241
[a11] O.H. Kegel, "Über den Normalisator von subnormalen und erreichbaren Untergruppen" Math. Ann. , 163 (1966) pp. 248–258
[a12] J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Oxford (1987)
[a13] E. Ormerod, "The Wielandt subgroup of a metacyclic $p$-group" Bull. Austral. Math. Soc. , 42 (1990) pp. 499–510
[a14] D.J.S. Robinson, "Groups in which normality is a transitive relation" Proc. Cambridge Philos. Soc. , 60 (1964) pp. 21–38
[a15] D.J.S. Robinson, "On the theory of subnormal subgroups" Math. Z. , 89 (1965) pp. 30–51
[a16] J.E. Roseblade, "On certain subnormal coalition classes" J. Algebra , 1 (1964) pp. 132–138
[a17] E. Schenkman, "On the norm of a group" Illinois J. Math. , 4 (1960) pp. 150–152
[a18] H. Wielandt, "Über den Normalisator der subnormalen Untergruppen" Math. Z. , 69 (1958) pp. 463–465
How to Cite This Entry:
Wielandt subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wielandt_subgroup&oldid=18195
This article was adapted from an original article by Derek J.S. Robinson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article