A course in abstract algebra, Edition: version 17 Jun 2016 by Nicholas Jackson

By Nicholas Jackson

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1 Let G = ( G, ∗) be a group, let H ⊆ G be a nonempty subset of G, and let ∗ H be the induced operation obtained by restricting ∗ to H × H. 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.

64, page 32. 4 (1986) 195–215 • H Wussing, The Genesis of the Abstract Group Concept, Dover (2007) Two overviews of the history of group theory from the late 18th century to the early 20th century. Exercises Let ∗ be a binary operation defined on R such that x ∗ y = x + y + xy. (a) Is ∗ associative? (b) Is ∗ commutative? (c) Is there an element e ∈ R which serves as an identity with respect to ∗? (d) Does every element x ∈ R have an inverse x −1 with respect to ∗? 2 Let S = { a, b} be a set with two elements.

56 Let σ = ( x1 x2 ) be a period–2 cyclic permutation in some (possibly infinite) symmetric group Sym( X ). Then we call σ a transposition. We noted earlier that σ3 = ι and τ 6 = ι. More formally, the order |σ | of σ in S5 is 3, while the order |τ | of τ in S5 is 6. The next proposition shows that we can easily work out the order of a permutation π ∈ Sn just by looking at its disjoint cycle representation. 57 Let π ∈ Sn be a permutation in some (possibly infinite) symmetric group Sym( X ). Then if π can be represented as a finite product π = σ1 σ2 .

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