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A course in abstract algebra by Nicholas Jackson

By Nicholas Jackson

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Example text

Conversely, suppose that HK = KH. To show that it is a subgroup of G, we need to confirm that it is closed under inverses and multiplication. For the first of these, suppose that h ∈ H and k ∈ K, so that hk ∈ HK. Then (hk)−1 = k−1 h−1 ∈ KH = HK, so HK contains all required inverses. Now suppose that h1 , h2 ∈ H and k1 , k2 ∈ K. Then (h1 k1 )(h2 k2 ) = h1 (k1 h2 )k2 .

Then H = ( H, ∗ H ) is a subgroup of G (written H < G) if and only if: SG1 H is closed under the action of ∗ H . That is, for all h1 , h2 ∈ H, the product h1 ∗ H h2 ∈ H too. SG2 For all h ∈ H, the inverse h−1 ∈ H as well. 2 Let G = ( G, ∗) be abelian, and let H < G be a subgroup of G. Then H is also abelian. The converse doesn’t hold, however, since nonabelian groups can have abelian subgroups. For example, we noted earlier that D3 has a subgroup isomorphic to Z3 which is therefore abelian, but D3 itself isn’t abelian.

4 Show that the multiplication table for a finite group G satisfies the Latin square property. That is, show that each element of the group occurs exactly once in each row and column of the table. 5 Let ∗ : R×R → R by x ∗ y = xy + 1. Is this operation commutative? Does it determine a group structure on R? If so, prove it by verifying the necessary criteria; if not, which axioms fail? 6 Which of the following are groups? For any that are not, say which of the axioms G0–G3 fail. +c , for a, b, c, d ∈ Z and b, d > 0 (a) (Z, ×) (e) (Q, ∗) where ba ∗ dc = ba+ d (b) (Z3 , ×3 ) (f) (Zeven , +) (c) (Z4 , ×4 ) (g) (Zodd , +) (d) (Zn , ×n ) if n isn’t prime (h) (Z, −) Here, × represents the usual multiplication operation defined on Z, and ×n represents multiplication modulo n.

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