In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {W i } i∈I for a Hilbert space H , there exists a Hilbert space K ⊇ H and an orthonormal basis of subspaces {N i } i∈I for K such that W i = P (N i ), where P is the orthogonal projection of K onto H . We introduce a new definition of atomic resolution of the identity in Hilbert spaces. In particular, we define an atomic resolution operator for an atomic resolution of the identity, which even yield a reconstruction formula.

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ( 2 G 2 (q)), where q = 3 2n+1 for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to 2 G 2 (q). We infer that if G is a finite group satisfying |G| = | 2 G 2 (q)| and Γ(G) = Γ( 2 G 2 (q)) then G ∼ = 2 G 2 (q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.

Fusion frames and g-frames were considered recently as generalizations of frames in Hilbert spaces, in this paper we generalize some of the known results in frame theory to fusion frames and g-frames. We obtain new g-frames by considering g-frames for its components. We also obtain some results about alternate dual g-frames, excess of g-frames and stability of g-frames under small perturbations.

Abstract.In this article, we study tensor product of Hilbert C * -modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F , and we get more results.For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K.

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Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut (J2) and Aut (McL), and the automorphism group of PSL(2, 2n) where n=2sare also uniquely determined by their order components. Also we discuss about the characterizability of Aut (PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.