properties of conjugacy

Let S be a nonempty subset of a group G.  When g is an element of G, a conjugatePlanetmathPlanetmath of S is the subset


We denote here

gSg-1:=Sg. (1)

If T is another nonempty subset and h another element of G, then it’s easily verified the formulae

  • (ST)g=SgTg

  • (Sg)h=Sgh

The conjugates Hg of a subgroupMathworldPlanetmathPlanetmath H of G are subgroups of G, since any mapping


is an automorphismPlanetmathPlanetmathPlanetmathPlanetmath (an inner automorphismMathworldPlanetmath) of G and the homomorphic image of group is always a group.

The notation (1) can be extended to

SggG:=SG (2)

where the angle parentheses express a generated subgroup.  SG is the least normal subgroupMathworldPlanetmath of G containing the subset S, and it is called the normal closurePlanetmathPlanetmathPlanetmath of S.

Title properties of conjugacy
Canonical name PropertiesOfConjugacy
Date of creation 2013-03-22 18:56:35
Last modified on 2013-03-22 18:56:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Topic
Classification msc 20A05
Related topic NormalClosure2
Related topic NonIsomorphicGroupsOfGivenOrder
Defines normal closure