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MAT H 3020 A: Homework 3 Due Friday Oct ober 12 at No on To subm it , use the assign me nt box for the course which is across from the Nort h 5th floor Ross
elevators. Late homework will not be accepted .
Solve the followi ng. Carefully justify any statements yo u make.
1. Let G = GL (2,lf 2 ) be t he set of invertible 2 x 2 matrices wit h entries in Z j mod 2.
(a ) List the elements of G . You may use 1 for [11 and 0 for [0], just keep in mind that
1+1
= o.
(b) Verify tha t G is a grou p under matrix mul t iplica ti on.
(c) Determine t he order of each element of G.
(d) Is G abelian?
(e) Show that G behaves like D 3 , i. e., fin d
2
= e , a T = T2(j.
a ,T
E
G such t ha t
2. Let G be a. group . Define a rela ti on on G by g
h - 1gh = g'.
~
g' provided there exists h E G such that
(a) P rove that t he relat ion
~
a
= e, T 3
is an equivalence relation.
(b) For G = D4 determine the equivalence classes of
~.
3. Prove tha t. no group can have exact ly two elements of order 2.
Note: You m ay discuss these questions but must not share solutions. Do your own
writing. Work that shows evidence of having bee n copied will receive 0 points.
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