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Each row and each column in the group multiplication table lists each of the group elements once and only once. From this, it follows that no two elements may be in the ...

Each of the sets forming a direct product is said to be a direct factor. A group G is said to be a direct factor of the group G^' if G^' is isomorphic to the group direct ...

Every closed three-manifold with finite fundamental group has a metric of constant positive scalar curvature, and hence is homeomorphic to a quotient S^3/Gamma, where Gamma ...

The free module of rank n over a nonzero unit ring R, usually denoted R^n, is the set of all sequences {a_1,a_2,...,a_n} that can be formed by picking n (not necessarily ...

The geometry of the Lie group R semidirect product with R^2, where R acts on R^2 by (t,(x,y))->(e^tx,e^(-t)y).

In the classical quasithin case of the quasithin theorem, if a group G does not have a "strongly embedded" subgroup, then G is a group of Lie-type in characteristic 2 of Lie ...

The conjecture that the Artin L-function of any n-dimensional complex representation of the Galois group of a finite extension of the rational numbers Q is an Artin ...

A subgroup H of an original group G has elements h_i. Let x be a fixed element of the original group G which is not a member of H. Then the transformation xh_ix^(-1), (i=1, ...

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp(v) is defined ...

The finite simple groups of Lie-type. They include four families of linear simple groups: PSL(n,q) (the projective special linear group), PSU(n,q) (the projective special ...