Questions and Answers of A First Course In Abstract Algebra

Let p be a prime number. Find the number of generators of the cyclic group Zpr, where r is an integer ≥ 1.
Generalizing Exercise 51, let S be any subset of a group G. a. Show that Hs = {x ∈ G| xs = sx for all s ∈ S} is a subgroup of G. b. In reference to part (a), the subgroup HG is the
Show that in a finite cyclic group G of order n, written multiplicatively, the equation xm = e has exactly m solutions x in G for each positive integer m that divides n.
Let H be a subgroup of a group G. For a, b ∈ G, let a ~ b if and only if ab-1 ∈ H. Show that ~ is an equivalence relation on G.
With reference to Exercise 53, what is the situation if 1 < m < n and m does not divide n?Data from ex. 53 Show that in a finite cyclic group G of order n, written multiplicatively,
For sets H and K, we define the intersection H ∩ K by H ∩ K = {x |x ∈ H and x ∈ K}. Show that if H ≤ G and K ≤ G, then H ∩ K ≤G. (Remember: ≤ denotes "is a subgroup of," not "is a
Prove that every cyclic group is abelian.
Show that Zp has no proper nontrivial subgroups if p is a prime number.
Let G be a group and let Gn = {gn | g ∈ G}. Under what hypothesis about G can we show that Gn is a subgroup of G?
Show that a group with no proper nontrivial subgroups is cyclic.
Let G be an abelian group and let H and K be finite cyclic subgroups with |H| = r and |K| = s. a. Show that if r and s are relatively prime, then G contains a cyclic subgroup of order
Let z1 = |z1 | (cosθ1 + i sinθ1) and z2 = |z2 |(cos θ2 + i sinθ2). Use the trigonometric identities to derive z1z2 = |z1 ||z2|[cos(θ1 + θ2) + i sin(θ1 + θ2)].
Concern the binary operation * defined on S = {a, b, c, d, e} by means of Table 2.26.Is * commutative? Why? 2.26 Table | a * a b C d e b c abc bcae cabba e b b a b d d e b d C e d d C
Compute the given arithmetic expression and give the answer in the form a + bi for a, b ∈ R.(4 - i) (5 + 3i)
Concern the binary operation * defined on S = {a, b, c, d, e} by means of Table 2.26.Compute (b * d) * c and b * (d * c). Can you say on the basis of this computation whether * is associative? 2.26
In Exercises, decide whether the object described is indeed a set (is well defined). Give an alternate description of each set.{n ∈ Z+ |n is a large number}
Determine whether the given map ∅ is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not?(Z, +) with (Z, +) where ∅(n) = n + 1 for n ∈ Z
Compute the given arithmetic expression and give the answer in the form a + bi for a, b ∈ R.(-i)35
In Exercises, describe the set by listing its elements.{m ∈ Z |m2 - m < 115}
Concern the binary operation * defined on S = {a, b, c, d, e} by means of Table 2.26.Compute (a* b) * c and a * (b * c). Can you say on the basis of this computations whether * is associative? 2.26
Determine whether the given map ∅ is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not?(Z, +) with (Z, +) where ∅(n) = 2n for n ∈ Z
Compute the given arithmetic expression and give the answer in the form a + bi for a, b ∈ R.i23
In Exercises, describe the set by listing its elements.{m ∈ Z |mn = 60 for some n ∈ Z}
Concern the binary operation * defined on S = {a, b, c, d, e} by means of Table 2.26.Compute b *d, c * c, and [(a* c) *e] *a. 2.26 Table ab * a a b b C d e C b d bc C b bcb ca e b b e
Determine whether the given map ∅ is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not?(Z, +) with (Z, +) where ∅(n) = -n for n ∈ Z
Compute the given arithmetic expression and give the answer in the form a + bi for a, b ∈ R.i4
In Exercises, describe the set by listing its elements.{m ∈ Z |m2 = 3}
What three things must we check to determine whether a function ∅: S → S' is an isomorphism of a binary structure (S, *) with (S', *')?
Compute the given arithmetic expression and give the answer in the form a + bi for a,b ∈ R.i3
In Exercises, describe the set by listing its elements.{x ∈ R I x2 = 3}

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