Order is twice an odd number implies subgroup of index two
Contents
Statement
Suppose is a finite group and the order of is , where is an odd number. Then, has a Subgroup of index two (?), i.e., a subgroup of order .
Note that since index two implies normal, the subgroup of order is in fact a normal subgroup, and is thus the Brauer core (?) of the whole group.
Related facts
Stronger facts
- Burnside's normal p-complement theorem is a significant generalization of this result, and states that if a -Sylow subgroup is in the center of its normalizer, it has a normal complement.
- Cyclic Sylow subgroup for least prime divisor has normal complement is a generalization of the stated result, and this generalization follows from Burnside's normal p-complement theorem and the fact that Cyclic normal Sylow subgroup for least prime divisor is central.
- Frobenius' normal p-complement theorem
For a complete list of normal p-complement theorems, refer:
Category:Normal p-complement theorems
Related facts about index two and least prime index
- Index two implies normal
- Subgroup of least prime index is normal: If is the least prime dividing the order of a finite group , any subgroup of index is normal.
- Normal of least prime order implies central
Facts used
- Cayley's theorem: Any group can be embedded inside the symmetric group , where an element acts by left multiplication.
- Cauchy's theorem: If a prime divides the order of a finite group , then has an element of order .
- Index satisfies transfer inequality: If are subgroups of , then .
Proof
Elementary proof
Given: A finite group of order , where is an odd integer.
To prove: has a subgroup of index two.
Proof: By fact (1), consider the embedding of as a subgroup of . Let be the alternating group on . By definition is a subgroup of index two in .
- contains an element, say , of order two: This follows from fact (2).
- , viewed as an element of , is an odd permutation. In other words, : The cycle decomposition consists of cycles of length two each, i.e., an odd number of cycles of even length. Thus, is an odd permutation.
- is a subgroup of index two in : By fact (3), has index either one or two in . However, the previous step shows that is not contained in , so is a proper subgroup of . Thus, is a subgroup of index two in .
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}