Paper 3, Section II, D

(a) Let $G$ be a group, and $N$ a subgroup of $G$. Define what it means for $N$ to be normal in $G$, and show that if $N$ is normal then $G / N$ naturally has the structure of a group.

(b) For each of (i)-(iii) below, give an example of a non-trivial finite group $G$ and non-trivial normal subgroup $N \leqslant G$ satisfying the stated properties.

(i) $G / N \times N \simeq G$.

(ii) There is no group homomorphism $G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity.

(iii) There is a group homomorphism $i: G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity, but the map

$G / N \times N \rightarrow G, \quad(g N, n) \mapsto i(g N) n$

is not a group homomorphism.

Show also that for any $N \leqslant G$ satisfying (iii), this map is always a bijection.

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