Assignment 2 – Due Friday, September 19
Turn this in at the start of recitation on Friday, September 19.
1. Read Sections 2.5–2.6 of Herstein. (Judson covers similar material in Chapters 3, 6, and 10.)
2. In recitation, you proved that the set of invertible n × n real matrices GLn (R) is a group. Let
SLn (R) = {M ∈ GLn (R) | det M = 1}.
Show that SLn (R) is a subgroup of GLn (R). Describe the right cosets of SLn (R) in GLn (R).
3. Let H be a subgroup of a group G, and suppose that g1 , g2 ∈ G. Show that if Hg1 = Hg2 , then
g1−1 H = g2−1 H.
4. Suppose that G is a finite group and a, b ∈ G.
(a) Suppose that o(a) = 3 and o(b) = 7. Show that the order of G must be at least 21.
(b) Suppose that o(a) = 6 and o(b) = 9. What’s the smallest that the order of G can be? Find a
cyclic group G and two elements a and b that satisfy this property.
5. Let a ∈ G. Show that an = e if and only if n is a multiple of o(a). (Hint: What’s the remainder of n
divided by o(a)?)
6. Herstein, p. 46: #1, 3
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