Question from book Thomas W. Hungerford - Abstrac xb My Questions | bartlebyO File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201.....Flash Player will no longer be supported after December 2020.Turn offLearn moreof 621-- A' Read aloudV DrawF HighlightO Erase224B. 21. Prove Theorem 7.7.22. Let G = {e, a, b} be a group of order 3. Write out the operation table for G.[Hint: Exercise 28 in Section 7.1.]23. Let G be a group with this property: If a, b, cɛG and ab = ca, then b = c.Prove that Gis abelian.24. If (ab)? = ab² for all a, b, ɛ G, prove that Gis abelian.25. Prove that G is abelian if and only if (ab)1 = a'b¬' for all a, bɛG.26. Prove thatevery nonabelian group G has order at least 6; hence, every groupof order 2, 3, 4, or 5 is abelian. [Hint: If a, beGand ab + ba, show that theelements of the subset H = {e, a, b, ab, ba} are all distinct. Show that eithera ¢ H or a = e; in the latter case, verify that aba £ H.]27. If every nonidentity element of G has order 2, prove that Gis abelian.[Hint: |a| = 2 if and only if a + e and a = a. Why?)28. If a EG, prove that |a| = la||.29. If a, b, ce G, prove that there is a unique element xEG such that axb = c.30. If a, be G, prove that Jab| = |ba|.31. (a) If a, be G and ab = ba, prove that (ab)- = e.(b) Show that part (a) may be false if ab + ba.32. If |G| is even, prove that G contains an element of order 2. [Hint: The identityelement is its own inverse. See the hint for Exercise 27.]33. Assume that a, bɛGand ab = ba. If Ja| and |b| are relatively prime, prove thatab has order la||b|. [Hint: See Exercise 31.]34. Suppose Ghas order 4, but contains no element of order 4.(a) Prove that no element of G has order 3. [Hint: If Ig] = 3, then G consistsof four distinct elements g, g², = e, d. Now gd must be one of these four11:25 AMEPICe Type here to searchAiEPIC5012/11/2020

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29. If a, b, ce G, prove that there is a unique element xe G such that axb = c.