Problem 3. Let G be a group.
a) Let [IOTA] e(x) = x-1 for all x in G.
i. Show that , € Sym(G).
ii. Suppose G has order n and let a = ca(2). What is the cycle type of c?
iii. Deduce that if G has even order then G has an element of order 2.
b) Fix g E G and define [RHO] p(x) = g.r for all x in G.
i. Show that P E Sym(G).
ii. Suppose G has order n and let b = o(g). What is the cycle type of p?
iii. Deduce that o(g) I o(G) without appealing to Lagrange's Theorem.
Problem 4. Fill in the following table for S6 like we did for Sa in Lecture 11.
Using the table, determine (d) for all d. What is exp(S6)?
Problem 5. Find explicit elements of S = Sym(N) with the following cycle types:
a) (1,1,1, ;0) b) 1) c) (1,1,1, 1) d) (bonus) (0,0,0, 00).
[Your answers must be unambiguous. That does not mean you need to come up with formulas
or piecewise definitions; you just need to make it clear what the output would be given any
positive integer input. For example, you can write out a small part of each permutation
(using two-line or cycle notation) and indicate the pattern or the rest of the definition
words. No proofs required. The bonus must be flawless to count as correct.]
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