Paper 3, Section II, E

Groups | Part IA, 2012

Let pp be a prime number, and aa an integer with 1ap11 \leqslant a \leqslant p-1. Let GG be the Cartesian product

G={(x,u)x{0,1,,p2},u{0,1,,p1}}G=\{(x, u) \mid x \in\{0,1, \ldots, p-2\}, u \in\{0,1, \ldots, p-1\}\}

Show that the binary operation

(x,u)(y,v)=(z,w)(x, u) *(y, v)=(z, w)


zx+y(modp1)wayu+v(modp)\begin{aligned} z & \equiv x+y(\bmod p-1) \\ w & \equiv a^{y} u+v(\bmod p) \end{aligned}

makes GG into a group. Show that GG is abelian if and only if a=1a=1.

Let HH and KK be the subsets

H={(x,0)x{0,1,,p2}},K={(0,u)u{0,1,,p1}}H=\{(x, 0) \mid x \in\{0,1, \ldots, p-2\}\}, \quad K=\{(0, u) \mid u \in\{0,1, \ldots, p-1\}\}

of GG. Show that KK is a normal subgroup of GG, and that HH is a subgroup which is normal if and only if a=1a=1.

Find a homomorphism from GG to another group whose kernel is KK.

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