Math 4441/6441 (Spring 2008) Homework Set #12 (Due 4/24) Assignment
For every integer mโฅ2, consider the quotient group of (Z,+) modulo the subgroup hmi,
i.e., Z/hmi, which we denote by Zm. (Note that the textbook denotes Z/hmiby Im.)
In particular, Zm={[0],[1],...,[mโ1]}, which is a group under โ+โ. Also, recall that
U(Zm) = {[r]โZm|gcd(r, m) = 1}, which is a group under multiplication.
Problem 1. Consider the quotient group Z8=Z/h8i(under addition) and also consider
the group U(Z8) (under multiplication).
(1) For each of the three elements x= [4],[5],[6] โZ8, determine ord(x) and hxi.
(2) Is Z8a cyclic group? If so, find a generator of Z8.
(3) Determine ord(u) for every uโU(Z8). Is U(Z8) a cyclic group? Why or why not?
Problem 2. Let Gbe a group and K๎G. For (fixed) elements a, b โG, consider aK and
bK in the quotient group G/K. (Prove either (1) or (2). One bonus point for both.)
(1) If (aK)(bK ) = (bK)(aK ) in G/K, show aโ1bโ1ab โK.
(2) Conversely, if aโ1bโ1ab โK, show (aK)(bK) = (bK )(aK) in G/K .
Problem 3. Let G, H be groups and f:GโHbe a group homomorphism.
(1) Let KโคGbe a (fixed) subgroup. Show f(K)โคH. (Recall f(K) = {f(k)|kโK}.)
(2) If K๎Gand fis onto, show f(K)๎H, i.e., f(K) is a normal subgroup of H.
Problem 4. Let Gbe a group, HโคGand K๎G(both fixed). Show KH โคG.
Problem 5 (6441 problem).Let Gbe a group and H, K โG.
(1) If HโคGand K๎G(both fixed), show HK =K H.
(2) True or false with reasoning: If H, K โคG, then KH =H K. .... True False
For the very same assignment with hints, click here.
Notice. All quoted results, such as Theorem x.y, are from the textbook A First Course in
Abstract Algebra with Applications (3rd Ed.) by J. J. Rotman, Pearson Prentice Hall.
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