Groups, Rings and Modules | Part IB, 2004

Let KK be a subgroup of a group GG. Prove that KK is normal if and only if there is a group HH and a homomorphism ϕ:GH\phi: G \rightarrow H such that

K={gG:ϕ(g)=1}K=\{g \in G: \phi(g)=1\}

Let GG be the group of all 2×22 \times 2 matrices (abcd)\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) with a,b,c,da, b, c, d in Z\mathbb{Z} and adbc=1a d-b c=1. Let pp be a prime number, and take KK to be the subset of GG consisting of all (abcd)\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) with ad1(modp)a \equiv d \equiv 1(\bmod p) and cb0(modp).c \equiv b \equiv 0(\bmod p) . Prove that KK is a normal subgroup of G.G .

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