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Every nontrivial finite -group has nontrivial center, in fact divides
Statement
If is a nontrivial finite -group, then
In particular, the center contains a nonidentity element.
Facts & Assumptions
Given: A nontrivial finite -group .
Nontriviality means with (A finite -group has order for a prime and some ).
For an action of on a finite set , (If a finite -group acts on a finite set , then ).
Conjugation gives a homomorphism (The map is a homomorphism with kernel and image ).
A homomorphism into a symmetric group defines an action (Actions of on correspond exactly to homomorphisms ).
The center is the set of elements commuting with every element of (The center of a group).
Congruence modulo means divisibility of the difference by (Congruence modulo an integer: when , including the moduli and ).
Proof
By [L3] and [L4], acts on itself by conjugation. An element is fixed by all conjugations exactly when it lies in by [L5].
Applying [L2] to this action gives .
By [L1], divides . Hence [L6] and step 2.1 show that divides . Since contains the identity and its cardinality is a positive multiple of , it also contains a nonidentity element.
Depends on
- A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in\mathbb N$
- If a finite $p$-group $P$ acts on a finite set $X$, then $|X|\equiv|X^P|\pmod p$
- The map $g\mapsto(x\mapsto gxg^{-1})$ is a homomorphism $G\to\operatorname{Aut}(G)$ with kernel $Z(G)$ and image $\operatorname{Inn}(G)$
- Actions of $G$ on $X$ correspond exactly to homomorphisms $G\to\operatorname{Sym}(X)$
- The center $Z(G)$ of a group
- Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$, including the moduli $0$ and $1$
Used by
- Every group of order p², for prime p, is abelian Corollary
- S₃ has trivial center, so the finite p-group hypothesis in the nontrivial-center theorem is necessary Counterexample
- Elementary groups are supersolvable Lemma
- For a finite p-group in characteristic p, the augmentation ideal of the group algebra is nilpotent Lemma
- Finite 2-transitive groups have affine or almost simple socle type Proposition
- A nonabelian group of order p³ is extraspecial Theorem
- Every finite p-group is nilpotent Theorem
- Schur-Zassenhaus existence theorem Theorem
Dependency tree · two levels
26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- K. Conrad, Group Actions, Theorem 5.1 (standard reference, not scraped)
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.2 (standard reference, not scraped)