Algebra and Geometry | Part IA, 2005

Let GG be a 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 G.G .

Let HH be the set of all 3×33 \times 3 real 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 invertible real matrices under multiplication.

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

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