A homomorphism from the group to the group of invertible linear transformations of a vector space.

A homomorphism from the group to the group of invertible linear transformations of a module.

A homomorphism from the group to the group of invertible linear transformations of a ring.

A homomorphism from the group to the group of invertible linear transformations of a field.

The number of elements in the group.

The number of generators of the group.

The number of conjugacy classes in the group.

The number of irreducible representations of the group.

The trace of the representation.

The determinant of the representation.

The eigenvalues of the representation.

The eigenvectors of the representation.

The inner product of two characters of a representation is zero if the representations are not equivalent.

The inner product of two characters of a representation is one if the representations are equivalent.

The inner product of two characters of a representation is equal to the number of conjugacy classes in the group.

The inner product of two characters of a representation is equal to the order of the group.

Every representation of a group is completely reducible.

Every representation of a group is semisimple.

Every representation of a group is irreducible.

Every representation of a group is trivial.

Every semisimple ring is isomorphic to a direct sum of matrix rings.

Every semisimple ring is isomorphic to a direct sum of division rings.

Every semisimple ring is isomorphic to a direct sum of fields.

Every semisimple ring is isomorphic to a direct sum of modules.

Every semisimple algebra is isomorphic to a direct sum of matrix algebras.

Every semisimple algebra is isomorphic to a direct sum of division algebras.

Every semisimple algebra is isomorphic to a direct sum of fields.

Every semisimple algebra is isomorphic to a direct sum of modules.

The largest nilpotent ideal of the ring.

The largest radical ideal of the ring.

The largest prime ideal of the ring.

The largest maximal ideal of the ring.

Every semisimple ring is a direct sum of simple rings.

Every semisimple ring is a direct sum of division rings.

Every semisimple ring is a direct sum of fields.

Every semisimple ring is a direct sum of modules.

Every semisimple module is a direct sum of simple modules.

Every semisimple module is a direct sum of indecomposable modules.

Every semisimple module is a direct sum of projective modules.

Every semisimple module is a direct sum of injective modules.

Every semisimple algebra is a direct sum of simple algebras.

Every semisimple algebra is a direct sum of division algebras.

Every semisimple algebra is a direct sum of fields.

Every semisimple algebra is a direct sum of modules.

Every semisimple group is a direct product of simple groups.

Every semisimple group is a direct product of division groups.

Every semisimple group is a direct product of fields.

Every semisimple group is a direct product of modules.

Every finite group of order $p^n$ is solvable.

Every finite group of order $p^n$ is nilpotent.

Every finite group of order $p^n$ is simple.

Every finite group of order $p^n$ is perfect.

Every finite group of odd order is solvable.

Every finite group of odd order is nilpotent.

Every finite group of odd order is simple.

Every finite group of odd order is perfect.

Every finite group is either solvable or semisimple.

Every finite group is either solvable or nilpotent.

Every finite group is either solvable or simple.

Every finite group is either solvable or perfect.