Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.
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please do f) only
![(d) Define * on Q by a * b = ab. Determine whether the binary operation * gives a group on
a given set. If no group results, write down the axiom that does not hold.
(e) Let S be the set of all real numbers except -1. Define * on S by a * b = a+b + ab. Show
that if * is a binary operation on a set S, then (S, *) is a group.[Hint: assume associativity, prove
all other axioms].
(f) Let G be a group and suppose that a * b * c = e. Show that b * c * a = e.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faceaba47-a7be-4966-b037-77aceb0ea16a%2F3992409d-7578-4bf3-9aaf-d18c8434abb8%2Fjqab44p_processed.jpeg&w=3840&q=75)
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- 27. Suppose that is a nonempty set that is closed under an associative binary operation and that the following two conditions hold: There exists a left identity in such that for all . Each has a left inverse in such that . Prove that is a group by showing that is in fact a two-sided identity for and that is a two-sided inverse of .Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .If a is an element of order m in a group G and ak=e, prove that m divides k.
- 42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .
- let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 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.15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
- Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.