Algebra Handout 2.5: More on Commutative Groups

نویسنده

  • PETE L. CLARK
چکیده

1. Reminder on quotient groups Let G be a group and H a subgroup of G. We have seen that the left cosets xH of H in G give a partition of G. Motivated by the case of quotients of rings by ideals, it is natural to consider the product operation on cosets. Recall that for any subsets S, T of G, by ST we mean {st | s ∈ S, t ∈ T }. If G is commutative, the product of two left cosets is another left coset: (xH)(yH) = xyHH = xyH. In fact, what we really used was that for all y ∈ G, yH = Hy. For an arbitrary group G, this is a property of the subgroup H, called normality. But it is clear – and will be good enough for us – that if G is commutative, all subgroups are normal. If G is a group and H is a normal subgroup, then the set of left cosets, denoted G/H, itself forms a group under the above product operation, called the quotient group of G by H. The map which assigns x ∈ G to its coset xH ∈ G/H is in fact a surjective group homomorphism q : G → G/H, called the quotient map (or in common jargon, the " natural map "), and its kernel is precisely the subgroup H. xK to f (x) ∈ G ′. This is well-defined, because if xK = x ′ K, then x ′ = xk for some k ∈ K, and then f (x ′) = f (x)f (k) = f (x) · e = f (x), since k is in the kernel of f. It is immediate to check that q(f) is a homomorphism of groups. Because f is surjective, for y ∈ G ′ there exists x ∈ G such that f (x) = y and then q(f)(xK) = y, so q(f) is surjective. Finally, if q(f)(xK) = e, then f (x) = e and x ∈ K, so xK = K is the identity element of G/K. In other words, a group G ′ is (isomorphic to) a quotient of a group G iff there exists a surjective group homomorphism from G to G ′. Corollary 2. If G and G ′ are finite groups such that there exists a surjective group

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تاریخ انتشار 2011