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A rational homomorphism phi:G->G^' defined over a field is called an isogeny when dimG=dimG^'. Two groups G and G^' are then called isogenous if there exists a third group ...

For any integers a_i with 1<=a_1<a_2<...<a_k<=n, the proportion of permutations in the symmetric group S_n whose cyclic decompositions contain no cycles of lengths a_1, a_2, ...

The composition quotient groups belonging to two composition series of a finite group G are, apart from their sequence, isomorphic in pairs. In other words, if I subset H_s ...

A short exact sequence of groups A, B, and C is given by two maps alpha:A->B and beta:B->C and is written 0->A->B->C->0. (1) Because it is an exact sequence, alpha is ...

Let G be a group of group order h and D be a set of k elements of G. If the set of differences d_i-d_j contains every nonzero element of G exactly lambda times, then D is a ...

A transformation x^'=Ax is unimodular if the determinant of the matrix A satisfies det(A)=+/-1. A necessary and sufficient condition that a linear transformation transform a ...

The identity element I (also denoted E, e, or 1) of a group or related mathematical structure S is the unique element such that Ia=aI=a for every element a in S. The symbol ...

The dimension d of any irreducible representation of a group G must be a divisor of the index of each maximal normal Abelian subgroup of G. Note that while Itô's theorem was ...

Let pi be a unitary representation of a group G on a separable Hilbert space, and let R(pi) be the smallest weakly closed algebra of bounded linear operators containing all ...

For elliptic curves over the rationals Q, the group of rational points is always finitely generated (i.e., there always exists a finite set of group generators). This theorem ...