Alperin [1] has recently introduced a fundamental method for conjugating from one p-Sylow subgroup Q of a finite group G to a second p-Sylow subgroup P in a series of steps. The importance of this is that it provides information concerning the conjugacy in G of subsets of P, that is, fusion of subsets of P. In certain situations it is possible to extend these results to the case of conjugate p-...

This is a toy example to illustrate some of the techniques of fusion and transfer as described in Gorenstein’s text Finite Groups. Finite groups with a unique subgroup of order p (p an odd prime) are classified in terms of a cyclic p′-extension of relatively simple combinatorial data. This note is intended to address a slightly misstated exercise in somewhat more detail and to be a toy example ...

this work is a continuation of [a‎. ‎o‎. ‎asar‎, ‎‎‎on infinitely generated groups whose proper subgroups are solvable‎, ‎{em j‎. ‎algebra}‎, ‎{bf 399} (2014) 870-886.]‎, ‎where it was shown‎ ‎that a perfect infinitely generated group whose proper subgroups‎ are solvable and in whose homomorphic images normal closures of ‎finitely generated subgroups are residually nilpotent is a fitting‎‎$p$-g...

WN ′ = W (N) N ′ denotes the corresponding Atkin–Lehner involution defined for Γ0(N). (If N ′ = 1, W1 means the identity operator.) Then we define the modular group Γ ∗ 0 (N) to be Γ ∗ 0 (N) = 〈Γ0(N) ∪ {WN ′}N ′ ‖N 〉, i.e., Γ ∗ 0 (N) is generated by Γ0(N) and {WN ′}N ′ ‖N . Then Γ ∗ 0 (N) is a normalizer of Γ0(N) in GL + 2 (Q) = {A ∈M2(Q) | detA > 0}. The factor group Γ ∗ 0 (N)/Γ0(N) is abelian...

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material about general linear algebraic groups before one c...

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

Oncogenic microRNAs (miRs) have emerged as diagnostic biomarkers and novel molecular targets for anti-cancer drug therapies. Real-time quantitative PCR (qPCR) is one of the most powerful techniques for analyzing miRs; however, the use of unsuitable normalizers might bias the results. Tumour heterogeneity makes even more difficult the selection of an adequate endogenous normalizer control. Here,...