The chief purpose of this paper is to find all pairs (G, θ) where G is a finite special p-group, and θ is an automorphism of G acting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class of p-groups, which we call special F-groups, where F is a finite field of characteristic p. We point out that if p is odd, then an F-group has exponent p. On the other hand, every special 2-group is also a special GF(2)-group.

Let G be a real Lie group, A a closed subgroup of G and B an analytic subgroup of G. Assume that B normalizes A and that AB is closed in G. Then our main result (Theorem 1) asserts that .

This result generalizes Lemma 2 in the paper [4], G. Hochschild has pointed out to me that the proof of that lemma given in [4] is not complete but that it can be easily completed.

Let R be an associative ring (not necessarily with identity).

R is a left П-ring if it has the following property: Let M be a finitely generated left R-module, N a submodule of M and ϕ:N→M an epimorphism. Then ϕ is an isomorphism.

This note originated in an attempt to generalize the assertion of Problem 164 which was proposed by Moser [2].

Let us first state our generalization.

Let A be a finite dimensional commutative and associative algebra with identity, over a field K. We assume also that A is generated by one element and consequently, isomorphic to a quotient algebra of the polynomial algebra K[X]. If A=K[a] and bi =fi(A), fi(X) ∊ K[X], 1≤i≤r we find necessary and sufficient conditions which should be satisfied by fi(X) in order that A = K[b 1, …, br ].

The result can be stated as a theorem about matrices. As a special case we obtain a recent result of Thompson [4].

In fact this last result was established earlier by Mirsky and Rado [3]. I am grateful to the referee for supplying this reference.

Let Sn, r be the semigroup generated by two elements a and b with two defining relations

(1)

By symmetry, we may assume that n ≤ r. In [l] and [2] it was shown that the semigroups

and

Let A = (aij) be an n × n complex matrix. The permanent of this matrix is

where the sum is taken over all permutations p of the set {1, …, n}.

In a recent paper [1] E. H. Lieb proved an interesting theorem (see below) which he applied to verify some conjectures of M. Marcus and M. Newman. The purpose of this note is to give a simple proof of Lieb's theorem.

J. A. Erdös proved recently [1] that every singular matrix over a field F is a product of idempotent matrices. He gave two proofs, one valid for matrices which are similar to triangular matrices and the other valid in general. We shall give a simple geometric proof of the above result. Instead of matrices we use linear operators. Moreover we get an explicit factorization in terms of projectors (idempotent operators).

Dans cet article nous considerons l'inégalité

où les xi désignent les nombres réels qui remplissent les conditions suivantes

Cette inégalité est proposée par H. S. Shapiro [1].