generated) abelian group that is not free abelian. One reason that is not free abelian is that it is divisible, meaning that, for every element and every nonzero integer , it is possible to express as a scalar multiple of another element . In contrast, non-trivial free abelian groups are never divisible, because in a free abelian group the basis elements cannot be expressed as multiples of other elements. Symmetry The symmetries of any group can be described as group automorphisms, the invertible homomorphisms from the group to itself. In non-abelian groups these are further subdivided into inner and outer automorphisms, but in abelian groups all non-identity automorphisms are outer. They form another group, the automorphism group of the given group, under the operation of composition. The automorphism group of a free abelian group of finite rank is the general linear group , which can be described concretely (for a specific basis of the free automorphism group) as the set of invertible integer matrices under the operation of matrix multiplication. Their action as symmetries on the free abelian group is just matrix-vector multiplication. The automorphism groups of two infinite-rank free abelian groups have the same first-order theories as each other, if and only if their ranks are equivalent cardinals from the point of view of second-order logic. This result depends on the structure of involutions of free abelian groups, the automorphisms that are their own inverse. Given a basis for a free abelian group, one can find involutions that map any set of disjoint pairs of basis elements to each other, or that negate any chosen subset of basis elements, leaving the other basis elements fixed. Conversely, for every involution of a free abelian group, one can find a basis of the group for which all basis elements are swapped in pairs, negated, or left unchanged by the involution. Relation to other groups If a free abelian group is a quotient of two groups , then is the direct sum . Given an arbitrary abelian group , there always exists a free abelian group and a surjective group homomorphism from to . One way of constructing a surjection onto a given group is to let be the free abelian group over , represented as formal sums. Then a surjection can be defined by mapping formal sums in to the corresponding sums of members of . That is, the surjection maps