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The monster group is the highest order sporadic group M. It has group order |M| = (1) = (2) where the divisors are precisely the 15 supersingular primes (Ogg 1980). The ...

Rubik's group is the group corresponding to possible rotations of a Rubik's Cube. There are six possible rotations, each corresponding to a generator of the group, and the ...

The trivial group, denoted E or <e>, sometimes also called the identity group, is the unique (up to isomorphism) group containing exactly one element e, the identity element. ...

A symmetry group is a group of symmetry-preserving operations, i.e., rotations, reflections, and inversions (Arfken 1985, p. 245).

Let K be a number field and let O be an order in K. Then the set of equivalence classes of invertible fractional ideals of O forms a multiplicative Abelian group called the ...

When p is a prime number, then a p-group is a group, all of whose elements have order some power of p. For a finite group, the equivalent definition is that the number of ...

The group of functions from an object G to itself which preserve the structure of the object, denoted Aut(G). The automorphism group of a group preserves the multiplication ...

The Heisenberg group H^n in n complex variables is the group of all (z,t) with z in C^n and t in R having multiplication (w,t)(z,t^')=(w+z,t+t^'+I[w^*z]) (1) where w^* is the ...

There are two definitions of a metacyclic group. 1. A metacyclic group is a group G such that both its commutator subgroup G^' and the quotient group G/G^' are cyclic (Rose ...

A permutation group is a finite group G whose elements are permutations of a given set and whose group operation is composition of permutations in G. Permutation groups have ...