The University of Manchester Algebra Math Calculus Exam Practice

PLEASE CIRCLE →THE NUMBERS OF
THE QUESTIONS
THAT YOU WOULD
LIKE ME
TO GRADE.
Question 1:
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Question 2:
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Question 3:
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Question 4:
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Question 5:
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1a)
Give an example of a group (G,*) and a non-empty subset H of G such that H is closed under the binary
operation * but is not a subgroup of G.
1b)
Is it possible in a group to have two different elements s and t such that s2 = t and t2 = s. Why or why not?
1c)
Prove that if (ab)2 = a2b2 for all elements in a group, then the group is commutative.
1d)
Let (G,●) be a commutative group whose elements are a1, a2, a3,….an.
Explain why the product (a1●a2●a3●…●an)2 must equal the identity of G.
2.
Definition: The square root of an element g in a group is an element x in the group such that x2 = g.
2a) Give the square root of each element in the group (Z5, +).
Kemeny’s Little Theorem: In any finite group of odd order, every element has a square root.
2b) Show that Kemeny’s Little Theorem would be false if the word “odd” were replaced by the word “even.”
2c) Use Kemeny’s Little Theorem to prove that in any finite group of odd order, only the identity is its own
inverse.
3)
Let S be the set of all real numbers except 1. (S,○) is a group if we define x○y = xy – (x+y) + 2.
3a)
What is the identity of this group? Why?
3b)
Prove that (S,○) is isomorphic to the set of all non-zero real numbers under multiplication.
4a)
Show an isomorphism between the group whose table is
given at the right and the group Z6.
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4b)
Let (G,*) and (H,◌) be groups. Use the mathematical definition of isomorphism to show that if (G,*) is
isomorphic to (H,◌) and φ is the isomorphism from G to H then φ maps the identity of G to the identity of H.
4c)
Let R+ represent the group of positive real numbers under multiplication. Is the mapping φ from R+ to R+ defined
by φ(𝑥) = √𝑥 an isomorphism? Explain.
5)
Canvas Guilano is staring at a partially destroyed 26-element table for a binary operation * defined on
S = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}. He was able to show that * is an associative operation and
that the cancellation laws hold for * immediately before most of the table was destroyed by battery acid. Now
Canvas is trying to finish the proof that (S,*) is a group. He can see only the first row of the table. Fortunately
that row is complete. Here it is:
|a b c d e f g h i j k l m n o p q r s t u v w x y z
a| p v w f q e z y x u t s r o m n l k j a b c d g h i
5a) Explain how Canvas can find an element that might be the group identity be using only the table’s first row.
5b) The identity must commute with every group element. Explain why the element you found in part (a) of this
problem will commute with the element “a.” (Recall that elements x and y are said to commute if xy = yx.)
5c) Prove that the element you found in part (a) of this problem is, in fact, an identity for the group.