Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem.
Every normal subgroup is the kernel of a group homomorphism and vice versa.
The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups.
This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set.
Multiplication of strings is defined by concatenation, for instance (abb) • (bca) = abbbca.
General linear group, denoted by GL(n, F), is the group of n-by-n invertible matrices, where the elements of the matrices are taken from a field F such as the real numbers or the complex numbers.