Consider the additive group
a.
b.
Sec.
Let
and
Where
Prove that
Prove that
For notational simplicity, write
As long as it is understood that the additions in
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Elements Of Modern Algebra
- 5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward
- Label each of the following statements as either true or false. Let x,y, and z be elements of a group G. Then (xyz)1=x1y1z1.arrow_forward45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )arrow_forwardLet G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.arrow_forward
- For 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_forward14. Let be an abelian group of order where and are relatively prime. If and , prove that .arrow_forward11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.arrow_forward
- Let be a subgroup of a group with . Prove that if and only if .arrow_forward9. Suppose that and are subgroups of the abelian group such that . Prove that .arrow_forwardLabel each of the following statements as either true or false. The Cayley table for a group will always be symmetric with respect to the diagonal from upper left to lower right.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,