Let G have order 2013. For p = 3, 11, 61 denote by np the number of Sylow p-groups in G. By Sylow’s theorems we have 61 | (n61 − 1) and n61|33, which is possible only for n61 = 1. Hence the 61-Sylow subgroup B is unique and therefore normal in G. Similarly, n11 | 3× 61 and 11 | (n11 − 1) yields n11 = 1; and the unique 11-Sylow subgroup A is normal in G. Note that A ∩ B is the trivial subgroup {...

THEOREM. Let G be a finite group and s an involution in G, such that CG(S)~CGQ(SO)Assume G^Co(s)0(G) (0(G) is the maximal normal odd order subgroup of G). Then Gc^Go. In particular, G has the following properties : (i) G has order 2 • 3 • 5 • 7 • 11 • 23, and is simple. (ii) G has two conjugacy classes of involutions. One class is represented by the involution s. A representative t of the secon...

Alperin [I] 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 Pin 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-s...

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-...

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.

let $h$, $l$ and $x$ be subgroups of a finite group$g$. then $h$ is said to be $x$-permutable with $l$ if for some$xin x$ we have $al^{x}=l^{x}a$. we say that $h$ is emph{$x$-quasipermutable } (emph{$x_{s}$-quasipermutable}, respectively) in $g$ provided $g$ has a subgroup$b$ such that $g=n_{g}(h)b$ and $h$ $x$-permutes with $b$ and with all subgroups (with all sylowsubgroups, respectively) $v$...

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...

A 2-Sylow subgroup of J is elementary abelian of order 8 and J has no subgroup of index 2. If r is an involution in J, then C(r) = (r) X K, where K _ A5. Let G be a finite group with the following properties: (a) S2-subgroups of G are abelian; (b) G has no subgroup of index 2; and (c) G contains an involution t such that 0(t) = (t) X F, where F A5. Then G is a (new) simple group isomorphic to J...