Exercise 36, § 6

Let $p$ be a prime number and let $G$ be a noncommutative group of order $p^3$. a) Now prove that, with the notations of the preceding exercise, one has $C = D$, 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 nonzero. Now prove that $G$ is semidirect product of a group of order $p$ by a cyclic group of order $p^2$. Now prove that there exist elements $x, y$ generating $G$ such that

that $G$ is characterized up to isomorphism by this property (and by the fact of being of order $p^3$), and that such a group $G$ exists. c) For $p = 2$, same question as in b) replacing the assumption $\theta \neq 0$ by the assumption that $G$ contains a noncentral element of order 2. Now prove that such a group is isomorphic to the dihedral group $D_4$ (cf. I, p. 134, exerc. 4) as well as 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. Now prove that there are elements $x, y, z$ generating $G$ such that that $G$ is characterized by this property, and that such a group exists. \* Now prove that $G$ is isomorphic to the multiplicative group of the matrices of the form whose elements $a, b, c$ belong to the field with $p$ elements.* e) Suppose that $p = 2$, and that no noncentral element of $G$ is of order 2 (therefore they are all of order 4). Now prove that $G$ is generated by elements $x, y$ such that that $G$ is characterized by this property, and is isomorphic to the quaternion group (cf. I, p. 134, exerc. 4). f) \* The group of matrices where $a, b, c$ run through the field with 2 elements, is it of type c) or of type e)? \*