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Math 455 Winter 2007 Test #1, Take 1
Each question except for #4 is worth 20 points.
#1) Make a subgroup diagram for Z8 .
#2) a. Is hR, −i a group? (In other words, do the real numbers form a group under
subtraction?) Justify your answer.
b. Let K be the set of all 6-digit sequences of 0’s and 1’s. For example: 001010 ∈ K. Define
a binary operation ∗ on K as follows: if a, b ∈ K, then the ith digit of a ∗ b is 0 if the ith
digit of a is the same as the ith digit of b; but if they are different, then the ith digit of a ∗ b
is 1. Is K an abelian group? Justify your answer. (You may assume that * is associative;
you do not need to justify this.)
(Example: a = 001010, b = 101110. Then a ∗ b = 100100, since: The first digit of a is 0.
The first digit of b is 1. They are different. So the first digit of a ∗ b is 1. The second digit
of a is 0, sameÃas the second digit of b. So!the second digit of a ∗ b is 0. Etc.)
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#3) Let τ =
be an element of S8 .
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a. Is τ odd or even? Justify your answer.
b. What is the order of τ ?
#4) Please make sure your cell phone is turned off.
#5) Cayley tables for three groups G, H, and K are given below. (You may assume that
they are groups; you do not need to prove this.)
a. Is G an abelian group? In one line, justify your answer.
b. In one or two sentences, explain how you know that G and H are not isomorphic. (You
do NOT need to prove your answer in detail.)
c. In one line, explain how you know that G is not a cyclic group.
d. Either H or K is a cyclic group. Which one? Justify your answer. (You do NOT need
to justify the fact that the other one is not cyclic.)
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#6) Let H = {σ ∈ S47 | σ(22) = 22}. Prove that H ≤ S47 .
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