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Exercise. Describe All Homomorphisms from Z24 to Z18. Solution

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Exercise. Describe All Homomorphisms from Z24 to Z18. Solution 1. Exercise. Describe all homomorphisms from Z24 to Z18. Solution. We need some preliminary discussion before we try to define the homomorphisms to try to figure out how they might look like. Let f : Z24 → Z18 be a homomorphism. For any integer k ≥ 0 notice that f(k mod 24) = f((1 mod 24)+ ··· +(1mod 24)) (added k times) = f(1 mod 24)+ ··· + f(1 mod 24) = k · f(1 mod 24). It follows that a homomorphism f is completely determined by the value f(1 mod 24). Write f(1 mod 24) = n mod 18 where n is an integer such that 0 ≤ n ≤ 17. Notice that 24n mod 18 = 24f(1 mod 24) = f(24 mod 24) = f(0 mod 24) = 0 mod 18. since any homomorphism maps the identity to identity. So 18 must divide 24n, so 3 must divide 4n, hence 3 must divide n. So n must be among 0, 3, 6, 9, 12, 15. The above discussion suggests how we might want to define our homomorphisms. We find that any such homomorphism must send (1 mod 24) to (3k mod 18) for some k =0, 1, ··· , 5. It follows that there are at most 6 possible homomorphisms from Z24 → Z18, given by f0, f1, ··· , f5 where fk(r mod 24) = 3kr mod18 (1) for k =0, 1, ··· , 5. First we need to check that for each of this k, equation (1) gives a well ′ ′ defined function from Z24 to Z18. To this end, let r, r ∈ Z such that r mod 24 = r mod 24. Then (r − r′) is a multiple of 24. So (3kr − 3kr′)=3k(r − r′) is a multiple of 3 × 24 = 72. In particular, (3kr − 3kr′) is a multiple of 18. It follows that (3kr mod 18) only depends on the congruence class of r modulo 24. So indeed equation (1) gives us a well defined function from Z24 to Z18. Now we verify that for any r, s ∈ Z, we have fk(r mod 24+ s mod 24) = fk((r + s)mod 24)= 3k(r + s) mod 18 =3kr mod 18+3ks mod 18 = fk(r mod 18) + fk(s mod 18). So each fk is indeed a homomorphism from Z24 to Z18. At the risk of repeating ourselves, we recall that if f : Z24 → Z18 is any homomorphism, then f(1 mod 18) must be of the form (3k mod 18) for some k = 0, 1, ··· , 5 and that any homomorphism f is determined completely by the value of f(1 mod 18). For k =0, 1, ··· , 5, we have constructed a homomorphism fk such that fk(1mod18) = 3k mod 18. It follows that there are six homomorphisms from Z24 to Z18 and these are given by f0, f1, ··· , f5 as defined in equation (1). 1 2. If m is an integer, then mZ denotes the set of all integer multiples of m. 2.1. Lemma. Let m, n be nonzero integers and r = lcm(m, n). Then mZ ∩ nZ = rZ. Proof. Let x ∈ mZ ∩ nZ. Then x is a multiple of m and x is a multiple of n. Hence x is a multiple of r. So x ∈ rZ. So mZ ∩ nZ ⊆ rZ. On the other hand r is a multiple of m and n, so r ∈ mZ ∩ nZ. So rZ ⊆ mZ ∩ nZ since mZ ∩ nZ is a subgroup of Z. 2.2. Lemma. Let G be a group and a be an element of G. Then hami = {ak : k ∈ mZ}. Proof. If k ∈ mZ, then k has the form k = mt for some integer t, so ak = amt ∈hami. Thus {ak : k ∈ mZ}⊆hami. On the other hand, if g ∈ hami, then g has the form g = ams for some integer s. Let k = ms. Then g = ak and k ∈ mZ. So g ∈{ak : k ∈ mZ}. 2.3. Lemma. Let G be a group and a ∈ G. Then show that hami∩hani = halcm(m,n)i. Proof. Write r = lcm(m, n). From the previous lemma, we have hami∩hani = {ak : k ∈ mZ}∩{ak : k ∈ nZ} = {ak : k ∈ mZ ∩ nZ} = {ak : k ∈ rZ} = hari. Here the first and the final equality follows from lemma 2.2 and the third equality follows from lemma 2.1. The above is a rather straight forward proof. On the next page you will find a more elaborate argument which also ends up giving a proof of the same statement in a much more long winded manner. But along the way we prove a bunch of things which are of some importance in themselves. 2 All through the following discussion, let G be a group and let a be an element of G of finite order |a| = d. If m is an integer, we shall write m′ = gcd(m, d). Recall from the text that |am| = d/m′. 2.4. Lemma. If m and n are relatively prime nonzero integers, then (mn)′ = m′n′. Proof. Note that m′ is a factor of m and n′ is a factor of n. Since m and n are relatively prime, so are m′ and n′. Since m′ and n′ each divide d, it follows that m′n′ divides d. Since (m/m′) and (n/n′) are each relatively prime to d, so is the product (m/m′)(n/n′). Writing mn =(m′n′)((m/m′)(n/n′)), we find that gcd(mn, d)= m′n′. 2.5. Lemma. Let k, n be nonzero integers. If k is a multiple of n, then haki⊆hani. Proof. We can write k = nt for some t ∈ Z. So ak = ant ∈hani. So haki⊆hani. ′ 2.6. Lemma. Let m is a nonzero integer, then hami = ham i. ′ Proof. Note that m is a multiple of m′, so lemma 2.5 implies am ∈ham i. Conversely, we can write m′ = αm + β|a| for some integers α, β ∈ Z. So ′ am = aαm+β|a| = aαm ∈hami. 2.7. Lemma. (a) If b ∈hai, then |b| divides |a|. (b)If k, m are nonzero integers such that ak ∈hami, then m′ | k′. Proof. (a) We can write b = at for some nonzero integer t. So |b| = |a|/t′ is a factor of |a|. (b) By part (a), we have |ak| divides |am|, that is, |a|/k′ divides |a|/m′. So m′ divides k′. 2.8. Lemma. Let m, n be relatively prime nonzero integers. Then hami∩hani = hamni. Proof. Write H = hami∩hani. By lemma 2.5, we have hamni⊆hami and hamni⊆hani. So hamni ⊆ H. Notice that H is a subgroup of the cyclic group hami, so H is cyclic. So H = haki for some nonzero integer k. Since ak ∈ hami and ak ∈ hani, lemma 2.7(b) implies that m′ | k′ and that n′ | k′. Since m′ and n′ are relatively prime, we have m′n′ | k′. So k′ is a multiple ′ ′ of m′n′ =(mn)′ (by lemma 2.4) and lemma 2.5 implies that hak i⊆ha(mn) i. Now we obtain ′ ′ H = haki = hak i⊆ha(mn) i = hamni where the second and the final equality follows from lemma 2.6. 2.9. Lemma. Let m and n be two nonzero integers and l = lcm(m, n). Then hami∩hani = hali. Proof. Since a−m and am generate the same subgroup, without loss, we may assume that m is a positive integer and similarly, that n is a positive integer. Let d = gcd(m, n). Let b = ad. Then am = bm/d and an = bn/d. Now we have 2 hami∩hani = hbm/di∩hbn/di = hbmn/d i = hamn/di = hali where the second equality follows from 2.8 since (m/d) and (n/d) are relatively prime and the last equality follows since mn = dl. 3 3. Sign of a permutation, even and odd permutations. 3.1. Definition. We want to define the “sign” of a permutation. We start with the polyno- mial P (x1, ··· , xn) in n variables x1, ··· , xn with n(n − 1)/2 distinct linear factors (xi − xj) for 1 ≤ i < j ≤ n. For example, if n = 4, then P (x1, x2, x3, x4)=(x1 − x2)(x1 − x3)(x1 − x4)(x2 − x3)(x2 − x4)(x3 − x4). Let α ∈ Sn be any permutation. Say that a pair (i, j) is a crossing pair for α if (i < j) −1 −1 1 2 ··· n and (α >α (j)), that is, when we represent α in the form α(1) α(2) ··· ··· α(n) then in the second row “i is sitting under a bigger number than j”, that is, “i is sitting to the right of j”. Let N(α) be set of crossing pairs for α N(α)= {(i, j): 1 ≤ i < j ≤ n, α−1(i) >α−1(j)}. The number |N(α)| is called the number of crossings of α. Claim: The polynomial P (xα(1), ··· , xα(n)) also has n(n − 1)/2 distinct linear factors. Upto a plus or minus sign, these factors are same as the factors of P (x1, ··· , xn). More precisely, for each 1 ≤ i < j ≤ n, either (xi − xj) or (xj − xi) is a factor of P (xα(1), ··· , xα(n)), but not both. If (i, j) ∈/ N(α), then (xi − xj) is a factor, otherwise (xj − xi) is a factor. proof of claim: Pick a pair i, j such that 1 ≤ i < j ≤ n. Suppose (i, j) ∈/ N(α). Then −1 −1 α (i) <α (j), so (xα−1(i) −xα−1(j)) is a factor of P (x1, ··· , xn). So(xα(α−1(i)) −xα(α−1 (j)))= (xi − xj) is a factor of P (xα(1), ··· , xα(n)).
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