(a)
The quaternion group is a famous noncommutative group; we will define the group as the set
Show that the group is noncommutative
(b)
The quaternion group is a famous noncommutative group; we will define the group as the set
Which element of the group is identity element?
(c)
The quaternion group is a famous noncommutative group; we will define the group as the set
Find the inverse of each element in the group
(d)
The quaternion group is a famous noncommutative group; we will define the group as the set
Show that
(e)
The quaternion group is a famous noncommutative group; we will define the group as the set
Show that the set
(f)
The quaternion group is a famous noncommutative group; we will define the group as the set
Find the subgroup that has two elements.
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- Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)arrow_forwardFor an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication. (Sec. 3.5,3,6, Sec. 4.6,17). Find an isomorphism from the additive group 4={ [ 0 ]4,[ 1 ]4,[ 2 ]4,[ 3 ]4 } to the multiplicative group of units U5={ [ 1 ]5,[ 2 ]5,[ 3 ]5,[ 4 ]5 }5. Find an isomorphism from the additive group 6={ [ a ]6 } to the multiplicative group of units U7={ [ a ]77[ a ]7[ 0 ]7 }. Repeat Exercise 14 where G is the multiplicative group of units U20 and G is the cyclic group of order 4. That is, G={ [ 1 ],[ 3 ],[ 7 ],[ 9 ],[ 11 ],[ 13 ],[ 17 ],[ 19 ] }, G= a =e,a,a2,a3 Define :GG by ([ 1 ])=([ 11 ])=e ([ 3 ])=([ 13 ])=a ([ 9 ])=([ 19 ])=a2 ([ 7 ])=([ 17 ])=a3.arrow_forwardLet H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?arrow_forward
- (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.arrow_forward42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .arrow_forward40. Find the commutator subgroup of each of the following groups. a. The quaternion group . b. The symmetric group .arrow_forward
- Let be a subgroup of a group with . Prove that if and only if .arrow_forwardConstruct a multiplication table for the group D5 of rigid motions of a regular pentagon with vertices 1,2,3,4,5.arrow_forwardA subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,