finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra. Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian. In fact, for every prime number there are (up to isomorphism) exactly two groups of order , namely and . Classification The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when G has zero rank; this in turn admits numerous further generalizations. The classification was proven by Leopold Kronecker in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms by Carl Friedrich Gauss in 1801; see history for details. The cyclic group of order is isomorphic to the direct sum of and if and only if and are coprime. It follows that any finite abelian group is isomorphic to a direct sum of the form in either of the following canonical ways: the numbers are powers of (not necessarily distinct) primes, or divides , which divides , and so on up to . For example, can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: . The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic. For another example, every abelian group of order 8 is isomorphic to either (the integers 0 to 7 under addition modulo 8), (the odd integers 1 to 15 under multiplication modulo 16), or . See also list of small groups for finite abelian groups of order 30 or less. Automorphisms One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group . To