LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The center of a group is a normal subgroup

Statement

For every group $G$ , the center $Z (G)$ is a normal subgroup of $G$ .

Facts & Assumptions

Given: A group $G$ with identity $e$ and center $Z (G)$ .

[F1]

The center is $Z (G) = {z \in G : z g = g z for every g \in G}$ (The center $Z (G)$ of a group).

[F2]

A subset of a group is a subgroup when it contains the identity and is closed under products and inverses (Subgroup).

[L1]

A subgroup $N \leq G$ is normal if $g N g^{- 1} = N$ for every $g \in G$ (Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).

Proof

technique · direct

1.1

The identity lies in $Z (G)$ . If $x, y \in Z (G)$ and $g \in G$ , then $(x y) g = x (y g) = x (g y) = (x g) y = (g x) y = g (x y)$ , so $x y \in Z (G)$ . If $x \in Z (G)$ , then $x g = g x$ implies $x^{- 1} g = g x^{- 1}$ after multiplying by $x^{- 1}$ on both sides, so $x^{- 1} \in Z (G)$ . Hence $Z (G) \leq G$ .

F1F2algebra

1.2

If $z \in Z (G)$ and $g \in G$ , then $g z g^{- 1} = z g g^{- 1} = z$ . Therefore $g Z (G) g^{- 1} = Z (G)$ for every $g \in G$ .

F1algebra

2.1

Steps 1.1 and 1.2 show that $Z (G)$ is a subgroup invariant under conjugation, so it is normal.

step 1.1step 1.2L1∎

Depends on

Used by

Dependency tree · two levels

9 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

Encyclopedia of Mathematics, Characteristic subgroup (standard reference, not scraped)