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A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. This definition is ...

For every even dimension 2n, the symplectic group Sp(2n) is the group of 2n×2n matrices which preserve a nondegenerate antisymmetric bilinear form omega, i.e., a symplectic ...

A finite group L is quasisimple if L=[L,L] and L/Z(L) is a simple group.

In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group G acts on a set X (this process is called a group action), ...

X is a p^'-group if p does not divide the group order of X.

When the group order h of a finite group is a prime number, there is only one possible group of group order h. Furthermore, the group is cyclic.

The group algebra K[G], where K is a field and G a group with the operation *, is the set of all linear combinations of finitely many elements of G with coefficients in K, ...

The double covering group of the (linear) symplectic group.

The set of all nonsingular affine transformations of a translation in space constitutes a group known as the affine group. The affine group contains the full linear group and ...

The group of rotations and translations.