Exercise 36, § 6

Let p be a prime number and G a non-commutative group of order $p^3$.

(a) Show that, with the notation of the preceding exercise, C = D and that G/D is isomorphic to the product of two groups of order p.

(b) Suppose that p is odd and that the homomorphism $\theta$ of the preceding exercise is non-zero. Show that G is the semi-direct product of a group of order p by a cyclic group of order $p^2$. Show that there exist elements x, y generating G such that

that G is characterized up to isomorphism by this property (and by the fact that it is of order $p^3$) and that such a group G exists.

(c) For $p = 2$, consider the same question as in (b) with the hypothesis $\theta \neq 0$ replaced by the hypothesis that G contains a non-central element of order 2. Show that such a group is isomorphic to the dihedral group $\mathbf{D}_4$ (cf. Exercise 4) and also to a Sylow 2-subgroup of $\mathfrak{S}_4$.

(d) Suppose that $p$ is odd and that the homomorphism $\theta$ of the preceding exercise is zero. Show that there are elements $x, y, z$ generating $G$ such that

that $G$ is characterized by this property and that such a group exists. Show that $G$ is isomorphic to the multiplicative group of matrices of the form

$$\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{pmatrix}$$

with coefficients $a, b, c$ in the field with $p$ elements.

(e) Suppose that $p = 2$ and that no non-central element of $G$ is of order 2 (hence that they are all of order 4). Show that $G$ is generated by elements $x, y$ such that

that $G$ is characterized by this property and is isomorphic to the quaternionic group (cf. Exercise 4).

(f) Is the group of matrices

where $a, b, c$ run through the field with 2 elements, of type (c) or of type (e)?