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An extension of a group H by a group N is a group G with a normal subgroup M such that M=N and G/M=H. This information can be encoded into a short exact sequence of groups ...

A presentation of a group is a description of a set I and a subset R of the free group F(I) generated by I, written <(x_i)_(i in I)|(r)_(r in R)>, where r=1 (the identity ...

The upper central series of a group G is the sequence of groups (each term normal in the term following it) 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=... that is constructed in the ...

Given a number field K, there exists a unique maximal unramified Abelian extension L of K which contains all other unramified Abelian extensions of K. This finite field ...

If a subgroup H of G has a group representation phi:H×W->W, then there is a unique induced representation of G on a vector space V. The original space W is contained in V, ...

The Patterson graph is the graph whose vertices are the 22880 centers of the Sylow 3-groups of the Suzuki group and in which two vertices are adjacent whenever they generate ...

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 ...

Some elements of a group G acting on a space X may fix a point x. These group elements form a subgroup called the isotropy group, defined by G_x={g in G:gx=x}. For example, ...

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 icosahedral group I_h is the group of symmetries of the icosahedron and dodecahedron having order 120, equivalent to the group direct product A_5×Z_2 of the alternating ...