Order is twice an odd number implies subgroup of index two

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Statement

Suppose G is a finite group and the order of G is 2m, where m is an odd number. Then, G has a Subgroup of index two (?), i.e., a subgroup of order m.

Note that since index two implies normal, the subgroup of order m is in fact a normal subgroup, and is thus the Brauer core (?) of the whole group.

Related facts

Stronger facts

For a complete list of normal p-complement theorems, refer:

Category:Normal p-complement theorems

Related facts about index two and least prime index

Facts used

  1. Cayley's theorem: Any group G can be embedded inside the symmetric group \operatorname{Sym}(G), where an element g \in G acts by left multiplication.
  2. Cauchy's theorem: If a prime p divides the order of a finite group G, then G has an element of order p.
  3. Index satisfies transfer inequality: If A,B are subgroups of C, then [A:A \cap B] \le [C:B].

Proof

Elementary proof

Given: A finite group G of order 2m, where m is an odd integer.

To prove: G has a subgroup of index two.

Proof: By fact (1), consider the embedding of G as a subgroup of K = \operatorname{Sym}(G). Let L = \operatorname{Alt}(G) be the alternating group on G. By definition L is a subgroup of index two in G.

  1. G contains an element, say g, of order two: This follows from fact (2).
  2. g, viewed as an element of K = \operatorname{Sym}(G), is an odd permutation. In other words, g \notin L: The cycle decomposition consists of m cycles of length two each, i.e., an odd number of cycles of even length. Thus, g is an odd permutation.
  3. G \cap L is a subgroup of index two in G: By fact (3), G \cap L has index either one or two in G. However, the previous step shows that G is not contained in L, so G \cap L is a proper subgroup of G. Thus, G \cap L is a subgroup of index two in G.

Advanced proof

The statement is a particular case of the fact that cyclic Sylow subgroup for least prime divisor has normal complement, which in turn follows from Burnside's normal p-complement theorem.

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 122, Exercise 12, Section 4.2, More info