4 Definition Let G be a group and let H G Then H is a semico

4. Definition: Let G be a group and let H G. Then H is a semicom subgroup of G if for all a G and h H, there exists h H such that ah = ha. Let G be a group and let H G. Let G/H = {Ha : a G}. Define · on G/H as follows: if Ha,Hb G/H, then Ha·Hb = Hab. (d) Show that if G is a group and H is a semicom subgroup of G, then (G/H,·) is a group.

Solution

Notice that |H||K| is the size of H × K. Define a map f from H × K to HK by f : (h, k) hk. We shall show that for each x HK, the number of preimages of x in H × K is |H K|; this clearly implies the result. Suppose that f(h, k) = hk = x. Then for any g H K we have x = hgg1k = f(hg, g1k). It follows that for every element hg of h(H K) there is a preimage of x in H × K whose H coordinate is hg, and so there are at least |H K| preimages. Conversely, if f(h1, k1) is a preimage of x then hk = h1k1, and so h 1h1 = kk1 1 . Setting g = h 1h1, it is clear that g H K, and that h1 = hg. So all preimages of x have their H coordinate in h(H K), and so there are exactly |H K| of them

Notice that (hk) ¹ = k ¹h ¹ . Therefore for g G we have g HK g ¹ KH. It follows immediately that if HK is a subgroup then HK = KH. For the converse, suppose that HK = KH. Then certainly HK is non-empty, and contains inverses of its elements; so we need show only that it is closed under the group operation. But we have (h1k1)(h2k2) h1(KH)k2 = h1(HK)k2 = HK.

if G is a group and H is a subgroup of G, then (G/H,·) is a group.