Prove part
Theorem 3.4: Properties of Group Elements
Let
The identity element
For each
For each
Reverse order law: For any
Cancellation laws: If
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Chapter 3 Solutions
Elements Of Modern Algebra
- Prove part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.arrow_forward1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_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_forwardProve or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.arrow_forwardExercises 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_forward
- Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.arrow_forwardIf G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.arrow_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forward
- 5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forward9. Suppose that and are subgroups of the abelian group such that . Prove that .arrow_forwardLet H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,