Suzuki-2-group

A finite non-Abelian $2$-group $U$, other than the group of quaternions, which admits a cyclic group of automorphisms $\langle a\rangle$ that acts transitively on the set $\Omega$ of elements of order 2 of $U$. This means that for any two elements $x$ and $y$ of $\Omega$ there is a natural number $n$ such that $y=x^{a^n}$. In the Suzuki $2$-group $U$, the set $\Omega$ and the identity element constitute a subgroup $Z$ that coincides with the centre of $U$; the quotient group $U/Z$ is then elementary Abelian. If the order of $Z$ is equal to $q$, then the order of $U$ is equal to $q^2$ or $q^3$.

Suzuki $2$-groups have been fully described (see [1]). The name derives from the fact that in a Suzuki group, the Sylow $2$-group $U$ has these properties.

References

[1]	G. Higman, "Suzuki 2-groups" Ill. J. Math. , 7 : 1 (1963) pp. 79–96 Zbl 0112.02107