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Normal subgroup is expressible as union of conjugacy classes

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Let G be a group, and let H be a subgroup of G. Prove that H is a normal subgroup if and only if H can be expressed as the union of conjugacy classes of G.

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Solution Summary

A detailed proof of the given assertion (that a subgroup H of a given group G is a normal subgroup if and only if H can be expressed as the union of conjugacy classes of G) is presented. The definitions of normal subgroup and conjugacy class are reviewed.

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For every element g of G, the conjugacy class of g is the set C_g = {x*g*x^(-1): x in G}.

Let H be a subgroup of G.

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(i) First, we prove that if H is a normal subgroup, then H is the union of conjugacy classes of G.

So assume that H is a normal subgroup.

By the definition of normal subgroup, we have that, for every h in H and every x in G, x*h*[x^{-1}] is an element of H. Thus for every h in H, the conjugacy class C_h is a subset of H.

Note that, for every h in H, h is an element of C_h. (To see this, let x = e, the ...

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Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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