Math 5310, Exam 2, 23 September Name:
1. (8 points) Circle the co
ect answer. No justification needed for these.
(a) Groups are cool true false
(b) If g is an element of a group and gn = 1, then g has order n. true false
(c) If every element g in a group G satisfies g4 = 1 then G is commutative. true false
(d) Using cycle notation in S6, XXXXXXXXXX5) = XXXXXXXXXXtrue false
2. (4 points) Give an example a group homomorphism j ∶R+�→R× such that j(2) = 4.
(Write down your ideas, I will give credit for partial answers and good ideas)
3. (8 points) Let G be a group with a trivial centre, i.e. Z(G) = {1}. Prove that if g ≠h then cg ≠ch, that is, prove that if g
and h are different elements of G, then conjugation by g is not the same as conjugation by h.
(Write down your ideas, I will give credit for partial answers and good ideas)
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SOLUTIONS
0 Oo
0
Use 4K)=2×
.
This is a
homomorphism since
2×-19
=
2×29
and 20=1
gH : if Cg -- ch then
gxg-t-hxh-ifhallx.ua
ea
ange : x g-
'
=
-g' hxh "
so : Xj 'h= g-thx for all ×
So
g-
' he ZCG ) -- fi }
This means g-
'hi
,
and so g- h . It
(extra space: clearly label anything here)
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