Math 2.0

Group theory and generalizations Definition

Commutator of two elements

Builds on Group

Definition

Let GG be a group and let g,h∈Gg, h \in G. The commutator of gg and hh is the element [g,h]=g−1h−1gh[g, h] = g^{-1} h^{-1} g h of GG.

Remarks

The equation gh=hg[g,h]gh = hg [g, h] holds, so [g,h]=e[g, h] = e if and only if gh=hggh = hg. Some authors use the convention [g,h]=ghg−1h−1[g, h] = g h g^{-1} h^{-1} instead. The inverse of [g,h][g, h] is [h,g][h, g].

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Further reading

  1. Web https://en.wikipedia.org/wiki/Commutator, Group theory
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