We relate the connection between the sizes of circuits in suborbital graph for the normalizer of [Formula: see text] in PSL(2,[Formula: see text]) and the congruence equations arising from related group action. We give a number theoretic result which says that all prime divisors of [Formula: see text] for any integer u must be congruent to [Formula: see text].

In this study, firstly, we interpret the level set, support, kernel for bipolar complex fuzzy (BCF) characteristic function, and BCF point. Then, subgroup, normal conjugate, normalizer cosets, abelian factor group. Furthermore, present associated examples theorems, prove these theorems. After that, image pre-image of subgroups under homomorphism related

In this paper, we prove the p-nilpotency of a finite group with assumption that some subgroups of Sylow subgroup are weakly s-semipermutable subgroups in the normalizer of Sylow subgroups. Our results unify and generalize some earlier results. Mathematics Subject Classification: 20D10, 20D15

Let P be an extraspecial p-group which is neither dihedral of order 8, nor of odd order p and exponent p. Let G be a finite group having P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normalizer NG(P ).

Let p be a prime, k be a field of characteristic 6= p and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension edk(N ; p) of N at p for every n ≥ 1.

Let g be a complex simple Lie algebra. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The nilpotent radical of b is denoted by u. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The theory of ad-nilpotent ideals has attracted much recent attention in the work of Kostant, Cellini-Papi, Sommers, an...

For a group G, embedded in its of permutations B=Perm(G) via the left regular representation λ:G→B, normalizer λ(G) B is Hol(G), holomorph G. The set ℋ(G) those N≤Hol(G) such that N≅G and NormB(N)=Hol(G) keyed to structure so-called multiple NHol(G)=NormB(Hol(G)), conjugates by NHol(G). We wish generalize this considering certain

1. The universal p-Frattini cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. The p-Frattini module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Restriction to the normalizer of a p-Sylow. . . . . . . . . . . . . . . . . . . . . . . 8 4. Asymptotics of the p-Frattini modules Mn . . . . . . . . . . . . . . . . . . . . ....