7. The free 2-generated group in the quaternion group variety (for discussion in lecture if time permits)

Let $F = F _ V (2)$ for $V =$   Var$(D _ {8}) =$   Var$(Q _ {8})$.

Laws determining $V$ are $x ^ 4 = e$, $x ^ 2 y = y x ^ 2$.

Let $a,b$ be generators of $F$ and let $c = (ab) ^ {-1}$.

Every element of $F$ has the form $a ^ i b ^ j a ^ {2k} b ^ {2 \ell} c ^ {2m}$, where $0 \leq i,j,k,\ell,m \leq 1$.

$F$ is the semidirect product of    Z$_ 2 \times$   Z$_ 4$ by    Z$_ 4$ via powers of $\sigma(u,v) = (u+v,v)$.

See Figure .

Figure: Con$(F)$, the lattice of normal subgroups of $F$