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...

The articles [2], [3], [1], [4], and [5] provide the terminology and notation for this paper. For simplicity, we adopt the following rules: G denotes a group, A, B denote non empty subsets of G, N , H, H1, H2 denote subgroups of G, and x, a, b denote elements of G. Next we state a number of propositions: (1) For every normal subgroup N of G and for all elements x1, x2 of G holds x1 ·N · (x2 ·N)...

We study the subgroup of k -automorphisms of k ⁢ [ x , y ] which commute with a simple derivation d of k ⁢ [ x , y ] . We prove, for instance, that this subgroup is trivial when d is a shamsuddin simple derivation. in the general case of simple derivations, we obtain properties for the elements of this subgroup.

Let G be a classical algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using geometric arguments about the action of X on subspaces of the natural module for G. Infiniteness is proven using a dimension criterion which involves root systems. 1 Statement of results Let G be a classical algebraic group ...

For an affine algebraic variety X, we study the subgroup Autalg(X) of group regular automorphisms Aut(X) X generated by all connected subgroups. We prove that is nested, i.e., a direct limit subgroups Aut(X), if and only

Given a $p$-group $G$ and subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing subject to further properties. We show in Theorem I that one of the said is always $\equiv 1 \pmod p$ if only same holds for others. In II supplement above result by focusing on normal III obtain sharpened version c...

It is known that, if G is a triply transitive permutation group on a finite set X with a regular normal subgroup N, then |N| = 2 (d ̂ 2) or |N| = 3. (See [12; Theorem 11.3].) If N is a regular normal subgroup of a permutation group G on X, xeX, and Gx is the stabiliser of x, then Gx £ G/JV (as abstract group), and so G has a representation on a set Y such that, for xeX, the representations of Gx...

In previous papers [4; 5; 6] we constructed and studied many classes of operations on groups which have the following properties: (1) Associative; (2) Commutative; (3) The resultant group contains isomorphic images of the original factors and is generated by them; (4) In the resultant group the intersection of a factor with normal subgroup generated by the remaining factors is the unit element....

Glauberman’s classical Z∗-theorem is a theorem about involutions of finite groups (i.e. elements of order 2). It is one of the important ingredients for the classification of finite simple groups, which in turn allows to prove the corresponding theorem for elements of arbitrary prime order p. Let us recall the statement: if G is a finite group with a Sylow p-subgroup P , and if x is an element ...