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Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1.
The smallest nonabelian simple group is the alternating group of order 60, and every simple group of order 60 is isomorphic to . [2] The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and every simple group of order 168 is isomorphic to PSL(2,7). [3] [4]
Quaternion group, Hamiltonian group (all subgroups are normal without the group being abelian). The smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G. Extraspecial group. Dic 2, [3] Binary dihedral group <2,2,2>. [4] Nilpotent. 10 17 G 10 1: D 10: Z 5, Z 2 (5) Dihedral ...
Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order. [6]: 32 The concepts of abelian group and -module agree.
The solvable group 2 A 2 (2 2) is isomorphic to an extension of the order 8 quaternion group by an elementary abelian group of order 9. 2 A 2 (3 2) is isomorphic to the derived group G 2 (2)′. 2 A 3 (2 2) is isomorphic to B 2 (3). Remarks
Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; Hom(G,A) can be interpreted as an object of A.
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.
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.