0 Let G Ka, b, clai (a) Consider the element w abacba of G, so w can be viewed a
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Let G Ka, b, clai (a) Consider the element w abacba of G, so w can be viewed as a word with 6 letters. Prove that if we add a letter a, l, or c to the beginning or end of w, then when we apply appropriate relations from G we can reduce to a word that is shorter than 6 letters. b) Now, the "empty" word e has no letters, and so has length zero; it is the only length zero word in G. You may assume that there are exactly 3 distinct words of length 1 in G (namely the words a, b, and c) exactly 5 distinct words of length 2, exactly 6 distinct words of length 3, exactly 5 distinct words of length 4, exactly 3 distinct words of length 5, and exactly 1 word of length 6, for a total of 24 distinct words. Write down all of these 24 words. (c) A little while ago, in a series of exercises, we proved the following: With s1 (12) s2 (23), and sa (34), the symmetric group S. si, s2, sa (i e. Sit is generated y s1, 52, 83 moreover, we have 8 (s152)3 (s153)2 (s2s3) Assuming these foregoing facts, prove that G a S4.Explanation / Answer
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It's starting letter of whole numbers system.
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