In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Gorenstein–Walter theorem, proved by , states that if a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
''G'' has a
dihedral Sylow 2-subgroup
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...
, and ''O''(''G'') is the maximal
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of odd
order, then ''G''/''O''(''G'') is isomorphic to a
2-group, or the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
''A''
7, or a subgroup of PΓL
2(''q'') containing PSL
2(''q'') for ''q'' an odd prime power. Note that A
5 ≈ PSL
2(4) ≈ PSL
2(5) and A
6 ≈ PSL
2(9).
References
*
*
*
Theorems about finite groups
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