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For n>=3, there exist no additive finite and invariant measures for the group of displacements in R^n.

If a fixed point is added to each group of a special complete series, then the resulting series is complete.

A group of three elements, also called a triad. A triple is therefore a 3-tuple.

If a compact manifold M has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if M has negative curvature, then ...

A point group is a group of symmetry operations which all leave at least one point unmoved. Although an isolated object may have an arbitrary Schönflies symbol, the ...

A "split" extension G of groups N and F which contains a subgroup F^_ isomorphic to F with G=F^_N^_ and F^_ intersection N^_={e} (Ito 1987, p. 710). Then the semidirect ...

In an additive group G, the additive inverse of an element a is the element a^' such that a+a^'=a^'+a=0, where 0 is the additive identity of G. Usually, the additive inverse ...

An orbifold is the object obtained by identifying any two points of a map which are equivalent under some symmetry of the map's group.

Let j_k(alpha) denote the number of cycles of length k for a permutation alpha expressed as a product of disjoint cycles. The cycle index Z(X) of a permutation group X of ...

An edge automorphism of a graph G is a permutation of the edges of G that sends edges with common endpoint into edges with a common endpoint. The set of all edge ...