= Heisenberg group over a prime field
{c}
{title2=$UT_3(\mathbb F_p)$}
Upper-unitriangular three-by-three matrices over the prime <finite field> form the finite analogue of the real <Heisenberg group>. In coordinates their multiplication is $(a,b,c)(a',b',c')=(a+a',b+b',c+c'+ab')$. The order is $p^3$, and $(a,b,c)^k=(ka,kb,kc+\binom{k}{2}ab)$. For odd $p$, every nonidentity element has order $p$, but the group is nonabelian. For $p=2$ the exponent statement changes, so odd characteristic is essential for the <nonisomorphic Gassmann equivalent regular subgroups> construction.
Back to article page
