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| − | ''attainable subgroup''
| + | #REDIRECT [[Subnormal series]] |
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| − | Any member of any [[Subnormal series|subnormal series]] of a group. To indicate the subnormality of a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909701.png" /> in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909702.png" />, the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909703.png" /> is used.
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
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| − | ====Comments====
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| − | A subnormal subgroup is also called a subinvariant subgroup.
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| − | A subnormal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909704.png" /> that coincides with its commutator subgroup and whose quotient by its centre is simple is called a component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909705.png" />. The product of all components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909706.png" /> is known as the layer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909707.png" />. It is an important characteristic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090970/s0909708.png" /> in the theory of finite simple groups, see e.g. [[#References|[a1]]].
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Suzuki, "Group theory" , '''1–2''' , Springer (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Clarendon Press (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)</TD></TR></table>
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Latest revision as of 09:54, 3 January 2021
How to Cite This Entry:
Subnormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subnormal_subgroup&oldid=15071
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article