View source for Frattini subgroup

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201301.png" /> denote an arbitrary [[Group|group]] (finite or infinite) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201302.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201303.png" />) mean that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201304.png" /> is a subgroup (respectively, a normal subgroup) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201305.png" /> (cf. also [[Subgroup|Subgroup]]; [[Normal subgroup|Normal subgroup]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201306.png" /> denote a prime number.

The intersection of all (proper) maximal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201307.png" /> is called the Frattini subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201308.png" /> and will be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f1201309.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013011.png" /> is infinite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013012.png" /> may contain no maximal subgroups, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013013.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013014.png" />. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013015.png" /> is a characteristic (hence normal) subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013016.png" /> (cf. also [[Characteristic subgroup|Characteristic subgroup]]).

The set of non-generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013017.png" /> consists of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013018.png" /> satisfying the following property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013019.png" /> is a non-empty subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013022.png" />. In 1885, G. Frattini proved [[#References|[a1]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013023.png" /> is equal to the set of non-generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013024.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013025.png" /> is a [[Finite group|finite group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013026.png" /> for some subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013029.png" />. Using this observation, Frattini proved that the Frattini subgroup of a finite group is nilpotent (cf. also [[Nilpotent group|Nilpotent group]]). This basic result gave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013030.png" /> its name. Moreover, his proof was very elegant: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013031.png" /> denotes a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013032.png" />-subgroup (cf. also [[Sylow subgroup|Sylow subgroup]]; [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013033.png" />-group]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013034.png" />, then he proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013035.png" />, whence, as remarked above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013036.png" /> and the nilpotency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013037.png" /> follows. Since then, the enormously useful result that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013038.png" /> is a finite group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013040.png" /> is a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013041.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013043.png" />, is usually referred to as the Frattini argument.

The Frattini subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013045.png" /> is strongly interrelated with the [[Commutator subgroup|commutator subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013046.png" />. In 1953, W. Gaschütz proved [[#References|[a2]]] that for every (possibly infinite) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013047.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013049.png" /> denotes the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013050.png" /> (cf. also [[Centre of a group|Centre of a group]]). The stronger condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013051.png" /> is equivalent to the property that all maximal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013052.png" /> are normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013053.png" />. It follows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013054.png" /> is a [[Nilpotent group|nilpotent group]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013056.png" /> is a finite group, then, as discovered by H. Wielandt, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013057.png" /> is nilpotent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013059.png" /> is a finite [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013060.png" />-group]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013062.png" /> is the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013063.png" /> generated by all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013064.png" />-th powers of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013065.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013066.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013067.png" />-group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013068.png" />. It follows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013069.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013070.png" />-group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013071.png" /> is equal to the intersection of all normal subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013072.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013073.png" /> with elementary Abelian quotient groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013074.png" />.

The Frattini quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013075.png" /> has also some important properties. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013076.png" /> is cyclic (cf. also [[Cyclic group|Cyclic group]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013077.png" /> is cyclic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013078.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013079.png" /> is nilpotent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013080.png" /> is nilpotent. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013081.png" /> is finite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013082.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013083.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013084.png" /> divides also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013085.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013086.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013087.png" />-group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013088.png" /> is elementary Abelian and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013089.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013090.png" />-automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013091.png" /> which induces the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013092.png" />, then, by a theorem of Burnside, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013093.png" /> is the identity automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013094.png" />.

Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013095.png" /> be a group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013096.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013097.png" />. The Burnside basis theorem states that any minimal generating set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013098.png" /> has the same cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f12013099.png" />, and by a theorem of Ph. Hall the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f120130100.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f120130101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f120130102.png" />.

General references for these and more specific results concerning the Frattini subgroup are [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Frattini,   "Intorno alla generazione dei gruppi di operazioni"  ''Rend. Atti Acad. Lincei'' , '''1''' :  4  (1885)  pp. 281–285; 455–457</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Gaschütz,   "Über die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120130/f120130103.png" />-Untergruppe endlicher Gruppen"  ''Math. Z.'' , '''58'''  (1953)  pp. 160–170</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''I''' , Springer  (1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J.S. Robinson,   "A course in the theory of groups" , Springer  (1982)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.R. Scott,   "Group theory" , Prentice-Hall  (1964)</TD></TR></table>