For a given odd prime p, we investigate the power graphs of three classes of , nite groups: the elementary abelian groups of exponent p, and the extra special groups of exponents p or p. We show that these power graphs are Eulerian for every p. As a corollary, we describe two classes of non-isomorphic groups with isomorphic power graphs. In addition, we prove that the clique graphs of the power graphs of two considered classes are complete.

In [U. Dempwolff, On extensions of elementary abelian groups of order 25 by GL (5; 2), Rend. Sem. Mat. Univ. Padova, 48 (1972), 359 - 364.] Dempwolff proved the existence of a group of the form 25 GL (5; 2) (a non split extension of the elementary abelian group 25 by the general linear group GL (5; 2)).This group is the second largest maximal subgroup of the sporadic Thompson simple group Th: In this paper we calculate the Fischer matrices of Dempwolff group  G‾=25GL (5; 2): The theory of projective characters is involved and we have computed the Schur multiplier together with a projective character table of an inertia factor group. The full character table of G‾ is then can be calculated easily.

If T is a semi-abelian algebraic theory, we prove that the category BornT of bornological T-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in BornT and, for a bornological T-algebra X, we investigate the relation between the representability of actions on X as a T-algebra and as a bornological T-algebra. We investigate further the algebraic coherence and the algebraic local cartesian closedness of BornT and prove in particular that both properties hold in the case of bornological groups.

We call a Cayley graph G= Cay (G, S) normal for G, if the right regular representation R (G) of G is normal in the full automorphism group of Aut (G). In this paper, a classification of all non-normal Cayley graphs of finite abelian group with valency 6 was presented.

The number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving Gaussian binomial coefficients is presented.

Let A be an abelian category with enough projective objects and let X be a full subcategory of A. We define Gorenstein projective objects with respect to X and YX, respectively, where YX={YÎCh (A)|Y is acyclic and ZnYÎX}. We point out that under certain hypotheses, these two Gorensein projective objects are related in a nice way. In particular, if P (A) ÍX, we show that XÎCh (A) is Gorenstein projective with respect to YX if and only if Xi is Gorenstein projective with respect to X for each i, when X is a self-orthogonal class or X is Hom (-, X) -exact. Subsequently, we consider the relationships of Gorenstein projective dimensions between them. As an application, if A is of finite left Gorenstein projective global dimension with respect to X and contains an in- jective cogenerator, then we find a new model structure on Ch (A) by Hovey’s results in [14].