Homework 2, due 9/4/2015
1. Let m be a non-zero integer. Show that multiplication is well-defined on
Z/m, i.e. that if x ≡ x0 (mod m) and y ≡ y 0 (mod m), then xy ≡ x0 y 0
(mod m).
2. Let G be a group, and let X be any subset of G. In class we defined
the ‘subgroup generated by X,’ denoted hXi, to be the intersection of all
subgroups of G containing X. Show that hXi is in fact a subgroup. If we
let G be GLn (R), and let X be the subset of elementary matrices, what
is hXi?
3. Artin 2.2.1
4. Artin 2.2.4
5. Artin 2.9.4
6. Artin 2.9.6
7. (Bonus problem) Recall that the order of an element x in a group G is
the smallest positive integer n (if it exists–otherwise we say the order is
infinite) such that xn = 1. You might have wondered, if x and y are
elements of a group G with orders a and b, respectively, what can we say
about the order of xy? In general, nothing! To be precise: for any triple
of integers a, b, c > 1, there is a group G and elements x, y ∈ G such that
the orders of x, y, and xy are a, b, and c. Prove this in the special case
a = b = 2 (and c arbitrary); if you are feeling ambitious, prove the general
case.
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