Let
Where
Prove that
Prove that
For notational simplicity, write
As long as it is understood that the additions in
Sec.
Sec.
Sec.
Sec.
Prove or disprove that each of the following groups with addition as defined in Exercises
a.
b.
Sec.
Consider the additive group
a.
b.
Consider the additive group
a.
b.
Consider the additive groups
Let
Sec.
Consider the additive group
a.
b.
Sec.
Let
a. Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of
b. Prove that
c. Prove that
d. Give an example of rings
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Elements Of Modern Algebra
- 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_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forward5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forward
- Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forwardLet H and K be arbitrary groups and let HK denotes the Cartesian product of H and K: HK=(h,k)hHandkK Equality in HK is defined by (h,k)=(h,k) if and only if h=h and k=k. Multiplication in HK is defined by (h1,k1)(h2,k2)=(h1h2,k1k2). Prove that HK is a group. This group is called the external direct product of H and K. Suppose that e1 and e2 are the identity elements of H and K, respectively. Show that H=(h,e2)hH is a normal subgroup of HK that is isomorphic to H and, similarly, that K=(e1,k)kK is a normal subgroup isomorphic to K. Prove that HK/H is isomorphic to K and that HK/K is isomorphic to H.arrow_forward
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- 8. Consider the additive groups and . Define by . Prove that is a homomorphism and find ker . Is an epimorphism? Is a monomorphism?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_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
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,