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14E-b

Expert-verifiedFound in: Page 311

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**Let N be a normal subgroup of G , ${\mathit{a}}{\mathbf{\in}}{\mathit{G}}$ , and C the conjugacy class of a in G .**

**If ${{\mathit{C}}}_{{\mathbf{i}}}$ is any conjugacy class in G , prove that ${{\mathit{C}}}_{{\mathbf{i}}}{\mathbf{\subseteq}}{\mathit{N}}$ or ${{\mathit{C}}}_{{\mathbf{i}}}{\mathbf{\cap}}{\mathit{N}}{\mathbf{=}}{\mathit{\Phi}}$ .**

It is proved that, ${C}_{i}\subseteq N$ or ${C}_{i}\cap N=\Phi $** **.

It is given that N is a normal subgroup of G , $a\in G$ , C is the conjugacy class of a in G and ${C}_{i}$ is any conjugacy class in G .

If ** ${C}_{i}\cap N=\Phi $**, then there is nothing to prove.

If ${C}_{i}\cap N$ is non-empty, then part (a) implies that ${C}_{i}\subseteq N$ .

It has been concluded that, ${C}_{i}\subseteq N$ or ** ${C}_{i}\cap N=\Phi $**.

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