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Solution. The 11-Sylow subgroup is Z/11Z; the 5-Sylow subgroup is Z/5Z. By Sylow’s theorem, the 11-Sylow subgroup is normal. Hence, the group is a semi-direct product of its 5 and 11-Sylow subgroups. Since Aut(Z/11Z) = Z/10Z has a unique subgroup of order 5, there are up to isomorphism exactly two groups of order 55: the abelian group Z/55Z and the group with presentation ⟨x, y|x = y = 1, xyx−1...

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  is a finite group. then the set of all prime divisors of  is denoted by  and the set of element orders of  is denoted by . suppose that . then the number of elements of order  in  is denoted by  and the sizes of the set of elements with the same order is denoted by ; that is, . in this paper, we prove that if  is a group such that , where , then . here  denotes the family of suzuk...

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

First we will prove a small lemma. Lemma 1. Let P be a p-Sylow subgroup of G. Let N be a normal subgroup of G such that P ⊂ N . Then all p-Sylow subgroups are in N . Proof. Let P̃ be a p-Sylow subgroup that is not P . Then by the Sylow Theorems, we know that gPg−1 = P̃ for some g ∈ G. Since P ⊂ N , and N is normal, we know that gPg−1 ⊂ N and hence P̃ ⊂ N . Thus all p-Sylow subgroups are in N . Now...

These theories extend the existent proof of the first sylow theorem (written by Florian Kammueller and L. C. Paulson) by what is often called the second, third and fourth sylow theorem. These theorems state propositions about the number of Sylow p-subgroups of a group and the fact that they are conjugate to each other. The proofs make use of an implementation of group actions and their properties.