Paper 3, Section II, D

Groups | Part IA, 2015

(a) Let GG be a non-trivial group and let Z(G)={hG:gh=hgZ(G)=\{h \in G: g h=h g for all gG}g \in G\}. Show that Z(G)Z(G) is a normal subgroup of GG. If the order of GG is a power of a prime, show that Z(G)Z(G) is non-trivial.

(b) The Heisenberg group HH is the set of all 3×33 \times 3 matrices of the form

(1xy01z001)\left(\begin{array}{lll} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right)

with x,y,zRx, y, z \in \mathbb{R}. Show that HH is a subgroup of the group of non-singular real matrices under matrix multiplication.

Find Z(H)Z(H) and show that H/Z(H)H / Z(H) is isomorphic to R2\mathbb{R}^{2} under vector addition.

(c) For pp prime, the modular Heisenberg group HpH_{p} is defined as in (b), except that x,yx, y and zz now lie in the field of pp elements. Write down Hp\left|H_{p}\right|. Find both Z(Hp)Z\left(H_{p}\right) and Hp/Z(Hp)H_{p} / Z\left(H_{p}\right) in terms of generators and relations.

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