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Let X and Y be sets, and let R subset= X×Y be a relation on X×Y. Then R is a concurrent relation if and only if for any finite subset F of X, there exists a single element p ...

A set-theoretic term having a number of different meanings. Fraenkel (1953, p. 37) used the term as a synonym for "finite set." However, according to Russell's definition ...

Transitivity is a result of the symmetry in the group. A group G is called transitive if its group action (understood to be a subgroup of a permutation group on a set Omega) ...

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 ...

A set of generators (g_1,...,g_n) is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing ...

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, ...

A group action phi:G×X->X is called faithful if there are no group elements g (except the identity element) such that gx=x for all x in X. Equivalently, the map phi induces ...

The free product G*H of groups G and H is the set of elements of the form g_1h_1g_2h_2...g_rh_r, where g_i in G and h_i in H, with g_1 and h_r possibly equal to e, the ...

A module M is Noetherian if it obeys the ascending chain condition with respect to inclusion, i.e., if every set of increasing sequences of submodules eventually becomes ...

A nonempty set is a set containing one or more elements. Any set other than the empty set emptyset is therefore a nonempty set. Nonempty sets are sometimes also called ...