inverse element in a group

群的逆元

Algebra & Algebraic Geometry · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For $g$ in a group with identity $e$, the unique element $g^{-1}$ with $gg^{-1}=g^{-1}g=e$.

对群元素 $g$，与它在左右两侧运算均得到单位元的唯一元素。

Example

The additive inverse of $4$ in $\mathbb Z$ is $-4$.

Usage and distinctions

In a general monoid a one-sided inverse need not be a two-sided inverse.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. J. S. Milne — Group TheoryChapter 1, Definition 1.1, inverse axiom · Checked 2026-10-14

Translation sources

1. 李文威 — 代数学方法 卷一，作者源文件半群, 幺半群与群，items 左逆 and 右逆; following paragraph proving inverse uniqueness · Checked 2026-10-14

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