# finite abelian group in a sentence

- Clearly, every
*finite abelian group*is finitely generated. - This gave a
*finite abelian group*, as was recognised at the time. - In general, a
*finite abelian group*" G " is considered. - The automorphism group of a
*finite abelian group*can be described directly in terms of these invariants. - The families of
*finite Abelian groups*and finite nilpotent groups are almost full, but neither full nor Melnikov. - It's difficult to find
*finite abelian group*in a sentence. - It follows that any
*finite abelian group*" G " is isomorphic to a direct sum of the form - If G is a
*finite abelian group*, then G \ cong \ widehat { G } but this isomorphism is not canonical. - 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. - 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. - When " G " is a
*finite abelian group*, the group ring is commutative, and its structure is easy to express in terms of roots of unity. - The fundamental theorem of finitely generated abelian groups states that a finitely generated abelian group is the rank and a
*finite abelian group*, each of which are unique up to isomorphism. - reduced the principal ideal theorem to a question about
*finite abelian groups*: he showed that it would follow if the transfer from a finite group to its derived subgroup is trivial. - An arbitrary
*finite abelian group*is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. - Clearly Carmichael's theorem is related to Euler's theorem, because the exponent of a
*finite abelian group*must divide the order of the group, by elementary group theory. - Direct sums play an important role in the classification of abelian groups : according to the fundamental theorem of
*finite abelian groups*, every finite abelian group can be expressed as the direct sum of cyclic groups.

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