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Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form Z / p k Z {\displaystyle \mathbb {Z} /p^{k}\mathbb {Z} } for p {\displaystyle p} prime, and the latter ...
Isomorphisms: 2 B 2 (2) is the Frobenius group of order 20. Remarks: Suzuki group are Zassenhaus groups acting on sets of size (2 2n+1) 2 + 1, and have 4-dimensional representations over the field with 2 2n+1 elements. They are the only non-cyclic simple groups whose order is not divisible by 3. They are not related to the sporadic Suzuki group.
The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2.
Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups. [2] In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p. [4] (Note that in the finite case the direct product and direct sum coincide ...
An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z). Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group.
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. The automorphism group of a finite abelian group can be described directly in terms of these invariants.
It is the kernel of the signature group homomorphism sgn : S n → {1, −1} explained under symmetric group. The group A n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A 5 is the smallest non-abelian simple group, having order 60, and thus the smallest non-solvable group.