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The set of points of X fixed by a group action are called the group's set of fixed points, defined by {x:gx=x for all g in G}. In some cases, there may not be a group action, ...

The kernel of a group homomorphism f:G-->G^' is the set of all elements of G which are mapped to the identity element of G^'. The kernel is a normal subgroup of G, and always ...

Let Gamma be a representation for a group of group order h, then sum_(R)Gamma_i(R)_(mn)Gamma_j(R)_(m^'n^')^*=h/(sqrt(l_il_j))delta_(ij)delta_(mm^')delta_(nn^'). The proof is ...

A group representation of a group G on a vector space V can be restricted to a subgroup H. For example, the symmetric group on three letters has a representation phi on R^2 ...

If two groups are residual to a third, every group residual to one is residual to the other. The Gambier extension of this theorem states that if two groups are ...

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 ...

A general term which refers to an increase (or decrease in the case of the oxymoron "negative growth") in a given quantity.

If G is a perfect group, then the group center of the quotient group G/Z(G), where Z(G) is the group center of G, is the trivial group.

The problem of deciding if four colors are sufficient to color any map on a plane or sphere.

The gyroelongated pentagonal bicupola is the Johnson solid J_(46), which consists of two pentagonal cupolas, one twisted with respect to the other, adjoined to a decagonal ...