Consider the additive group
and the multiplicative group
and define
by
Prove that
Is
Find
and
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Elements Of Modern Algebra
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .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 a homomorphism from the group to the group . Prove part a of Theorem : If denotes the identity in and denotes the identity in , then . Prove part b of Theorem : for all in .arrow_forward24. Let be a group and its center. Prove or disprove that if is in, then and are in.arrow_forward
- Let H be a subgroup of the group G. Prove that if two right cosets Ha and Hb are not disjoint, then Ha=Hb. That is, the distinct right cosets of H in G form a partition of G.arrow_forward23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .arrow_forwardLet G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,