We establish and comment on a surprising relationship between the behaviour modulo a prime p of the number sn(G) of index n subgroups in a group G, and that of the corresponding subgroup numbers for a subnormal subgroup of p-power index in G. One of the applications of this result presented here concerns the explicit determination modulo p of sn(G) in the case when G is the fundamental group of...

An abundance of information regarding the structure of a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters — those induced from a linear character of some subgroup — since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the cl...

The theory of fuzzy sets with reference to subnormality needs to be properly explained. We shall in this article put forward the actual mathematical explanation of subnormal partial presence of an element in a set, better known as subnormal fuzziness. We shall first show that every law of subnormal fuzziness can be expressed in terms of two Lebesgue-Stieltjes measures. Thereafter we would defin...

The tactile analogue scale (TAS) was elaborated to be used in blind subjects or those who can not use the vision during their crises. The objective of this study was to characterize, from TAS, the architecture of migraine attacks in subjects with visual disability. For that, 11 migrainous with visual disturb (MVD) subjects were studied and 22 migrainous subjects with no visual disability as a c...

‎we pursue further our‎ ‎investigation‎, ‎begun in [h.~smith‎, ‎groups with all subgroups subnormal or‎ ‎nilpotent-by-{c}hernikov‎, ‎emph{rend‎. ‎sem‎. ‎mat‎. ‎univ‎. ‎padova}‎ ‎126 (2011)‎, ‎245--253] and continued in [g.~cutolo and h.~smith‎, ‎locally finite groups with all subgroups‎ ‎subnormal or nilpotent-by-{c}hernikov‎. ‎emph{centr‎. ‎eur‎. ‎j‎. ‎math.} (to appear)] ‎‎‎of groups $g$ in w...

Let $ G$ be a finite group and $p$ prime. \mathrm{Vo}(G) denote the set of orders vanishing elements, $\mathrm{Vo}_{p} (G)$ subset consisting those elements divisible by $\mathrm{Vo}_{p'} (G) not $p$. Dolfi, Pacifi, Sanus Spiga proved that if is p $-power for all a\in \mathrm{Vo}(G)$, then G has normal Sylow $-subgroup. In another article, same authors also show \mathrm{Vo}_{p'}(G) =\emptyset $...

This remarkable and brilliant book throws a searchlight on to the clinical study, treatment and training of a section of our community whose problems have been so misunderstood and ignored. The author's approach reveals much profound and original thinking based on a new concept of the condition which has been labelled "subnormality", and his opinions and findings are the result of years of expe...

A subgroup Q of a group G is commensurated if the commensurator of Q in G is the entire group G. Our main result is that a finitely generated group G containing an infinite, finitely generated, commensurated subgroup H, of infinite index in G is 1-ended and semistable at ∞. If additionally, Q and G are finitely presented and either Q is 1-ended or the pair (G,Q) has 1 filtered end, then G is si...

in this paper, we extend the construction of a fuzzy subgroup generated by a fuzzy subset to $l$-setting. this construction is illustrated by an example. we also prove that for an $l$-subset of a group, the subgroup generated by its level subset is the level subset of the subgroup generated by that $l$-subset provided the given $l$-subset possesses sup-property.