Centralizer product theorem
Contents
Statement
For faithful group actions
Suppose are distinct primes. Let be a finite -group, and be an abelian -group that is not cyclic. Suppose are the non-identity elements of , enumerated in any arbitrary order. Then, we have:
.
For general group actions
Suppose are distinct primes. Let be a finite -group, and be an abelian -group that is not cyclic. Suppose acts on by automorphisms. Suppose are the non-identity elements of , enumerated in any arbitrary order. Then, we have:
.
Note that the version for general group actions follows directly from the version for faithful group actions. In fact, for an action that is not faithful, it suffices to take the products of centralizers of coset representatives of the kernel of the action.
Related facts
Particular cases
Corollaries/applications
Facts used
- Centralizer product theorem for elementary abelian group
- Omega-1 of center is normality-large in nilpotent p-group (in fact, socle equals Omega-1 of center in nilpotent p-group)
- Central implies normal
Proof
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
This proof uses the principle of mathematical induction in a nontrivial way (i.e., it would be hard to write the proof clearly without explicitly using induction).
Given: Primes . A finite -group . An abelian non-cyclic subgroup . are the non-identity elements of , enumerated in any order.
To prove: .
Proof: We prove this by induction on the order of , combined with fact (1). Fact (1) settles the case for an elementary abelian -group.
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Let ; in other words, is the subgroup of comprising the identity element and elements of order in the center of . Then, acts on by automorphisms, and is a nontrivial elementary abelian -group. | Fact (2) | is a finite -group | [SHOW MORE] | |
2 | There exists a non-identity such that is nontrivial | Fact (1) | is a finite abelian non-cyclic -group, . | Step (1) | [SHOW MORE] |
3 | With selected as in Step (2), is a normal subgroup of | Fact (3) | Steps (1), (2) | [SHOW MORE] | |
4 | stabilizes and hence acts on | is abelian | [SHOW MORE] | ||
5 | Let . Then, | inductive hypothesis | Steps (3),(4) | [SHOW MORE] | |
6 | Steps (2)-(5) | [SHOW MORE] |
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 188, Theorem 3.16, Chapter 5, Section 3 (p'-automorphisms of p-groups), ^{More info}