Near Frattini subgroups of residually finite generalized free products of groups

LetG=A★HBbe the generalized free product of the groupsAandBwith the amalgamated subgroupH. Also, letλ(G)andψ(G)represent the lower near Frattini subgroup and the near Frattini subgroup ofG, respectively. IfGis finitely generated and residually finite, then we show thatψ(G)≤H, providedHsatisfies a nontrivial identical relation. Also, we prove that ifGis residually finite, thenλ(G)≤H, provided: (i)Hsatisfies a nontrivial identical relation andA,Bpossess proper subgroupsA1,B1of finite index containingH; (ii) neitherAnorBlies in the variety generated byH; (iii)H<A1≤AandH<B1≤B, whereA1andB1each satisfies a nontrivial identical relation; (iv)His nilpotent.

Congruence-free regular semigroups

A semigroup is said to be congruence-free if and only if its only congruences are the universal relation and the identical relation. Congruence-free inverse semigroups were studied by Baird [2], Trotter [19], Munn [15,16] and Reilly [18]. In addition, results on congruence-free regular semigroups have been obtained by Trotter [20], Hall [4] and Howie [7].

On a set of normal subgroups

The commutator [a, b] of two elements a and b in a group G satisfies the identityab = ba[a, b].The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation(A) [g, x]Y = [g, y]−1[g, xy]holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that{g}G = {g, K(g)}.

On a conjecture of Nagata

In a recent paper (1) Nagata proved that a (linear associative) algebra, not necessarily of finite dimension, over a field of characteristic 0 which satisfies the identical relation xn = 0 satisfies also the relation x1x2 … xN = 0, where N is an integer depending only on n. He remarked further that it is a corollary that the result remains true if the ground field is of prime characteristic p, provided that p is large enough compared with n; and he conjectured that the obviously necessary condition p > n is in fact sufficient. The object of this note is to prove Nagata's conjecture. To do this, we give a new proof of his theorem, and as a by-product we obtain a rather better bound for N than his, showing, namely, that we can take N = 2n − 1. The determination of the best possible value of N, or even of its order of magnitude, seems not to be easy; at any rate, the best I have been able to do in the opposite direction is to show that for large n we cannot take N as small as n2/e2, where e is the base of natural logarithms.

XIV.—An Identical Relation connecting Seven Vectors

Among the most useful results of quaternion or vectorial algebra we may count the identities and transformations worked out by Hamilton and Tait and their all too few followers. It is not merely that a vector identity is equivalent to three scalar identities—a fact which aids us greatly to condense our calculations in respect to bulk. Yet more important is the greater fruitfulness of a vectorial relation in giving rise to derived relations which we would be less likely to perceive from a purely scalar analysis.

I.—The Law of Extensible Minors in Determinants

§1. As a preliminary to the establishment of the law in question, it is necessary to state and exemplify another law to which I have elsewhere directed attention, viz.,THE LAW OF COMPLEMENTARIES.To every general theorem which takes the form of an identical relation between a number of the minors of a determinant or between the determinant itself and a number of its minors, there corresponds another theorem derivable from the former by merely substituting for every minor its cofactor in the determinant, and then multiplying any term by such a power of the determinant as will make the terms of the same degree.