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Abstract Algebra

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Non-Abelian p-Groups of Small Order

ABSALG-LQMKYR

Let $G$ be a group with neutral element $e$. The group $G$ is defined to be a $p$-group, where $p\ge2$ is a prime number if all its elements have order a power of $p$.

In other words, for every $g\in G$, the smallest positive integer $k$ such that $g^k=e$ is of the form $k=p^s$, $s\ge0$.

If $G=\{e\}$, then $e^{p^0}=e$ for all $p$, and we exclude this case when we use the term non-trivial $p$-group.

You are allowed to assume the fact that, up to isomorphism, the only non-abelian groups of order at most $8$:

$S_3$ (the symmetric group​ on three​ letters)
$D_8$ (dihedral group of order $8$)
...and
${\mathbb H}$ (the group of order $8$ given by the quaternions, also denoted $Q$ and $Q_2$).

Recall that the quaternion group has presentation:

$ {\mathbb H} = \langle -1,i,j,k \mid (-1)^2 = 1, \;i^2 = j^2 = k^2 = ijk = -1 \rangle ​$

Answer the following questions with a number.