In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Gorenstein–Walter theorem, proved by , states that if a
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
''G'' has a
dihedral Sylow 2-subgroup, 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 ...
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 (mathematics), 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 ...
''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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