p-Groups

Prove that there are at most 5 groups of order \(2\cdot 7^2\) up to isomorphism.

Prove that there are no simple groups of order \(2^7\cdot 3^2\).

Let \(G\) be a finite \(p\)-group, for \(p\) a prime integer. Let \(e\) denote the identity element, and \(Z\) the center of \(G\). If \(N\ne \{e\}\) is normal, prove that \(N\cap Z\ne \{e\}\).

Let \(G\) be a \(p\)-group with \(|G|>p\). Show that

1. \(G\) has nontrivial center;
2. \(G\) has a normal subgroup of every order \(p^m<|G|\).

Let \(G\) be a finite \(p\)-group, with \(|G|>p\). Prove that the order of \(\operatorname {Aut}(G)\) is divisible by \(p\).

Let \(G\) be a group of order \(p^2q\), where \(p\) and \(q\) are distinct primes. Show that \(G\) has a normal Sylow subgroup.

Let \(G\) be a group of order \(p^n\), where \(p\) is a prime number and \(n>0\) is an integer. Prove that the center of \(G\) is not trivial.

Determine the groups of order \(1225=5^2\cdot 7^2\) up to isomorphism.

Let \(p\) be a prime integer. Let \(G\) be a finite \(p\)-group, with identity \(e\), and center \(Z\). If \(N\ne \{e\}\) is a normal subgroup of \(G\), prove that \(N\cap Z\ne \{e\}\).

Let \(p\) be a prime. Determine all groups of order \(p^2\).

Let \(G\) be a finite group and \(p\) a prime number such that \(|G|=p^n\) for some \(n>0\). Prove that \(G\) has nontrivial center and that for all integers \(0<m<n\), \(G\) has at least one normal subgroup of order \(p^m\).

Suppose \(G\) is a group of order \(p^n\) where \(p\) is a prime and \(n>1\). If \(A\) is the group of all automorphisms of \(G\), prove that \(p\) divides the order of \(A\).

Let \(G\) be a group of order \(p^n\), where \(p\) is a prime number and \(n>0\) is an integer. Show that the center of \(G\) is not trivial.

Let \(G\) be a nonabelian group of order \(p^3\) for a prime \(p\) and and let \(Z\) be its center. Show that \(G/Z\cong \mathbb {Z}/p\mathbb {Z}\oplus \mathbb {Z}/p\mathbb {Z}\).

Let \(G\) be a nonabelian group of order \(p^3\) where \(p\) is a prime. Prove that \(Z(G)\) and \(G'\) are equal and have order \(p\). [Here \(Z(G)\) is the center of \(G\) and \(G'\) is the commutator subgroup of \(G\), i.e., \(G'\) is generated by all elements of the form \(x^{-1}y^{-1}xy\) with \(x,y\in G\).]

Let \(p\) be a prime number, and let \(G\) be a finite group of order \(p^n\) acting on a set \(X\) consisting of \(r\) elements. Show that the number of points fixed by all elements of \(G\) is congruent to \(r\) modulo \(p\).

Let \(G\) be a finite group of order \(p^n\) acting on a set \(X\) consisting of \(k\) elements. Show that the number of points fixed by all elements of \(G\) is congruent to \(k\) modulo \(p\).