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Normal Subgroups and Factor Groups

Table of Contents

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1. Definition 1

  1. Normal Subgroups

    A subgroup \(H\) of a group \(G\) is \(\textbf{\textit{normal}}\) in \(G\) if \(gH = Hg\) for all \(g \in G\). That is, a normal subgroup of a group G is one in which the right and left cosets are precisely the same.

2. Theorem 1

Let \(G\) be a group and \(N\) be a subgroup of \(G\). Then the following statements are equivalent.

  1. The subgroup \(N\) is normal in \(G\).
  2. For all \(g \in G\), \(gNg^{-1} \in N\).
  3. For all \(g \in G\), \(gNg^{-1} = N\).

3. Exercises

Date: 2026-05-20 Wed 00:00