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By Nicholas Jackson

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32, page 16. 34, page 17. 36, page 18. 38, page 18. 35, page 17. 37, page 18. 39, page 18. 40, page 18. 42, page 19. 46, page 22. 43, page 21. 45, page 21. 47, page 22. 48, page 22. 50, page 24. 36 Matrices yield an important and rich class of groups, many of which are nonabelian. 44 Matrix groups led naturally to a discussion of geometric operations on vector spaces, and in particular we considered isometries: operations which preserve distance. 50 Composition of permutations is groups associative, there is an obvious identity permutation ι, and bijections are necessarily invertible.

54 Any permutation σ ∈ Sn can be written as a product of disjoint cyclic permutations. Because these cyclic permutations are disjoint, they are independent: they don’t interact with each other, in the sense that since they act on disjoint subsets of X = {1, 2, 3, 4, 5}, they don’t step on each others’ toes. Therefore, it doesn’t matter what order we apply them to the set X, and so the cycles (1 2 4) and (3 5) commute. ) A slight disadvantage with cycle notation is that it isn’t so obvious how to multiply permutations together.

Kn = ι . . ι = ι. i k. Then Therefore, π n = ι if and only if li |n for 1 i k. So in particular, n must be the smallest positive integer which divides each of the li , which is to say that it must be the lowest common multiple lcm(li , . . , lk ). So |σ | = l (σ ) = 3, but |τ | = lcm(2, 3) = 6. Now let’s look again at the permutation σ = (2 3 5). This has length 3, but is it possible to express it as a product of even shorter (but possibly not disjoint) cyclic permutations? 58 Any finite permutation σ in some (possibly infinite) symmetric group Sym( X ) can be expressed as a product of (not necessarily disjoint) transpositions.

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A course in abstract algebra by Nicholas Jackson


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