The object of this paper is to classify (up to inner automorphism) the primitive, maximal rank, reductive subalgebras of the (complex) exceptional Lie algebras. By primitive we mean that the subalgebras correspond to (possibly disconnected) maximal Lie subgroups. In [3], the corresponding classification for the (complex) classical Lie algebras was completed, as was the classification for the no...

Let A denote the alternation limit algebra, studied by Hopenwasser and Power, and by Poon, which is the closed direct limit of upper triangular matrix algebras determined by refinement embeddings of multiplicity rk and standard embeddings of multiplicity sk. It is shown that the quotient of the isometric automorphism group by the approximately inner automorphisms is the abelian group ZZ where d...

Public-key cryptosystems nowadays are mostly based on number theoretic problems like factorization (RSA) and the discrete logarithm problem (Elgamal). However, such systems can be broken with quantum computers by applying Shor’s algorithms [1] for solving both problems, factorization and discrete logarithm, in polynomial time. Hence there is a need for post-quantum cryptography, i.e., methods r...

AbstractLet W be a non-empty subset of a free group. The automorphism of a group G is said to be a marginal automorphism, if for all x in G,x^−1alpha(x) in W^*(G), where W^*(G) is the marginal subgroup of G.In this paper, we give necessary and sufficient condition for a purelynon-abelian p-group G, such that the set of all marginal automorphismsof G forms an elementary abelian p-group.

abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

let $gamma$ be a normal subgroup of the full automorphism group $aut(g)$ of a group $g$‎, ‎and assume that $inn(g)leq gamma$‎. ‎an endomorphism $sigma$ of $g$ is said to be {it $gamma$-central} if $sigma$ induces the the identity on the factor group $g/c_g(gamma)$‎. ‎clearly‎, ‎if $gamma=inn(g)$‎, ‎then a $gamma$-central endomorphism is a {it central} endomorphism‎. ‎in this article the conditi...

It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group coherently extended along any outgoing homomorphism. One is thus motivated to define a notion (categorical) automorphism in an arbitrary category, morphism, theory such forms part covariant isotropy. In this paper, we prove categorical category $$\textsf{Group...

suppose $gamma$ is a graph with $v(gamma) = { 1,2, cdots, p}$and $ mathcal{f} = {gamma_1,cdots, gamma_p} $ is a family ofgraphs such that $n_j = |v(gamma_j)|$, $1 leq j leq p$. define$lambda = gamma[gamma_1,cdots, gamma_p]$ to be a graph withvertex set $ v(lambda)=bigcup_{j=1}^pv(gamma_j)$ and edge set$e(lambda)=big(bigcup_{j=1}^pe(gamma_j)big)cupbig(bigcup_{ijine(gamma)}{uv;uin v(gamma_i),vin ...

We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips–Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown–Douglas–Fillmore. We also give a simpler and self-contained proof of the Phillips–Weaver result. ...