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The set of sums sum_(x)a_xx ranging over a multiplicative group and a_i are elements of a field with all but a finite number of a_i=0. Group rings are graded algebras.

Let (K,|·|) be a valuated field. The valuation group G is defined to be the set G={|x|:x in K,x!=0}, with the group operation being multiplication. It is a subgroup of the ...

The dihedral group D_2 is a point group that is isomorphic to the vierergruppe and the finite group C_2×C_2.

In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group G acts on a set X (this process is called a group action), ...

A group whose left Haar measure equals its right Haar measure.

The finite group T is one of the three non-Abelian groups of order 12 (out of a total of fives groups of order 12), the other two being the alternating group A_4 and the ...

The symmetric group S_n of degree n is the group of all permutations on n symbols. S_n is therefore a permutation group of order n! and contains as subgroups every group of ...

A group set is a set whose elements are acted on by a group. If the group G acts on the set S, then S is called a G-set. Let G be a group and let S be a G-set. Then for every ...

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The nth homotopy group of a topological space X is ...

A group in which any decreasing chain of distinct subgroups terminates after a finite number.