Which of the following is a nontrivial subgroup of Z3 x Z3 where Z3 is the additive group of integers modulo 3 consisting of {0,1,2} and the Cartesian product is the set of ordered pairs? a) {(0,0),(2,2)} b) {(0,0),(1,2),(2,1)} c) {(0,0),(1,1),(1,2)} d) {0,1} 2. Which of the following is a nontrivial subgroup of the quaternion group of order 8? This group is plus or minus 1, J, K, L where signs multiply as usual, 1 is a multiplicative identity, the squares of J, K, L are -1 and JK=L, KJ=-L, KL=J, LK=-J, LJ=K, JL=-K. The subgroup in question is cyclic. a) {J,-K,L} b) {1,-J,K,L} c) {1,L} d) {1,K,-1,-K} 3. For the following two functions, which of these 4 alternatives is correct? f(x)=x3 -x from the real numbers to itself and g(x)={(1,1),(2,3),(3,1)} from the set {1,2,3} to itself. a) both are one-to-one b) the first, but not the second, is one-to-one c) the second, but not the first, is one-to-one d) neither is one-to-one 4. Compute the composition (product) of these two permutations : ( 1 2 3 4 ) ( 3 4 2 1 ) and ( 1 2 3 4 ) ( 2 4 3 1 ) . Is it a) the identity b) ( 1 2 3 4 ) ( 4 1 2 3 ) c) ( 1 2 3 4 ) ( 1 2 3 4 ) d) ( 1 2 3 4 ) ( 2 1 4 3 ) 5. Determine whether this permutation given in cycle form is odd or even and what its order is (two answers). (2,5,3,1,6,8)(4,7,9) a) even, order is 3 b) odd, order is 3 c) odd, order is 6 d) even, order is 6
1. Which of the following is a nontrivial subgroup of Z3 x Z3 where Z3 is the additive group of integers modulo 3 consisting of {0,1,2} and the Cartesian product is the set of ordered pairs?
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a) {(0,0),(2,2)}
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b) {(0,0),(1,2),(2,1)}
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c) {(0,0),(1,1),(1,2)}
- d) {0,1}
2. Which of the following is a nontrivial subgroup of the quaternion group of order 8? This group is plus or minus 1, J, K, L where signs multiply as usual, 1 is a multiplicative identity, the squares of J, K, L are -1 and JK=L, KJ=-L, KL=J, LK=-J, LJ=K, JL=-K. The subgroup in question is cyclic.
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a) {J,-K,L}
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b) {1,-J,K,L}
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c) {1,L}
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d) {1,K,-1,-K}
3. For the following two functions, which of these 4 alternatives is correct? f(x)=x3 -x from the real numbers to itself and g(x)={(1,1),(2,3),(3,1)} from the set {1,2,3} to itself.
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a) both are one-to-one
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b) the first, but not the second, is one-to-one
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c) the second, but not the first, is one-to-one
- d) neither is one-to-one
4. Compute the composition (product) of these two permutations :
| ( | 1 | 2 | 3 | 4 | ) |
| ( | 3 | 4 | 2 | 1 | ) |
and
| ( | 1 | 2 | 3 | 4 | ) |
| ( | 2 | 4 | 3 | 1 | ) |
. Is it
- a) the identity
- b)
| ( | 1 | 2 | 3 | 4 | ) |
| ( | 4 | 1 | 2 | 3 | ) |
- c)
( 1 2 3 4 ) ( 1 2 3 4 ) - d)
( 1 2 3 4 ) ( 2 1 4 3 )
5. Determine whether this permutation given in cycle form is odd or even and what its order is (two answers). (2,5,3,1,6,8)(4,7,9)
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a) even, order is 3
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b) odd, order is 3
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c) odd, order is 6
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d) even, order is 6
Let H be the subgroup of the symmetric group of degree 3 which consists of the elements e, (1,2) in cycle form. Here e denotes the identity element. What are the order of H and the index of H in the symmetric group of degree 3? (Two answers, in order)
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a) 2,4
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b) 2,3
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c) 3,2
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d) 1,6
7. Suppose that in a group G, xyx-1 =z. Then by definition
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a) y,z are in the same conjugacy class
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b) x,z are in the same conjugacy class
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c) x,y are in the same conjugacy class
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d) z is the identity element
The centralizer of an element x of a group G is the subgroup of all elements z such that xz=zx. It follows from this and Lagrange's theorem that
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a) The centralizer always equals the cyclic subgroup generated by x.
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b) The order of the centralizer divides the number of elements in G.
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c) The centralizer is a normal subgroup
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d) The index of the centralizer is a prime number.
9. Suppose a square is represented with these 4 vertices A, B, C, D in order: (1,1),(1,-1),(-1,-1),(-1,1). Note that these go clockwise around it. Which of the following is not a reflection symmetry of the square?
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a) A,B,C,D go in order to D,A,B,C
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b) A,B,C,D go in order to D,C,B,A
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c) A,B,C,D go in order to B,A,D,C
- d) A,B,C,D go in order to A,D,C,B
10. The nonzero elements modulo a prime p form a multiplicative group of p-1 elements, since only 0 is excluded among the p congruence classes. It follows from this fact and Lagrange's theorem that
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a) Every group is cyclic
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b) The square of any element is congruent to 1 modulo p
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c) The cube of any element is congruent to 1 modulo p
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d) The p-1 power of any integer which is not a multiple of p is congruent to 1 modulo p
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