show that under complex multiplication, G={1,-1.i,-i} is an abelian group?
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show that under complex multiplication, G={1,-1.i,-i} is an abelian group?
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- 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 2.16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
- Find all homomorphic images of the quaternion group.Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.
- Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.