Moufang Loops of Odd order \(p^4q^3\)

Bulletin of the Malaysian Mathematical Sciences Society, Feb 2017

It is known that all Moufang loops of order \(p^4\) are associative if p is a prime greater than 3. Also, nonassociative Moufang loops of order \(p^5\) (for all primes p) and \(pq^3\) (for distinct odd primes p and q, with the necessary and sufficient condition \(q\equiv 1({\text{ mod }}\ p)\)) have been proved to exist. Consider a Moufang loop L of order \(p^{\alpha }q^{\beta }\) where p and q are odd primes with \(p<q\), \(q\not \equiv 1 ({\text{ mod }}\ p)\) and \(\alpha ,\beta \in {\mathbb {Z}}^+\). It has been proved that L is associative if \(\alpha \le 3\) and \(\beta \le 3\). In this paper, we extend this result to the case \(p>3\), \(\alpha \le 4\) and \(\beta \le 3\).

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Moufang Loops of Odd order \(p^4q^3\)

Moufang Loops of Odd order p4q3 Andrew Rajah 0 Lois Adewoye Ademola 0 0 School of Mathematical Sciences , Universiti Sains Malaysia (USM), 11800 Penang , Malaysia It is known that all Moufang loops of order p4 are associative if p is a prime greater than 3. Also, nonassociative Moufang loops of order p5 (for all primes p) and pq3 (for distinct odd primes p and q, with the necessary and sufficient condition q ≡ 1(mod p)) have been proved to exist. Consider a Moufang loop L of order pαqβ where p and q are odd primes with p < q, q ≡ 1(mod p) and α, β ∈ Z+. It has been proved that L is associative if α ≤ 3 and β ≤ 3. In this paper, we extend this result to the case p > 3, α ≤ 4 and β ≤ 3. Communicated by Miin Huey Ang. Moufang loop; Maximal subloop; Order; Nonassociative 1 Introduction A loop L , · is called a Moufang loop if it satisfies the identity (x · y) · (z · x ) = (x · (y · z)) · x . L , · being a loop has elements of the form u = (x · y) · z and v = x · (y · z) for all x , y, z ∈ L. L , · is not necessarily associative. So, if x , y and z are fixed elements of L, u and v may be identical or two different elements in L. Whatever B Lois Adewoye Ademola the case, there would exist a unique element k ∈ L such that u = v · k. (If u and v happen to be the same element, then obviously k = 1, the identity element). We use the notation k = (x , y, z) since k is nevertheless a function of these three elements, i.e., (x · y) · z = (x · (y · z)) · (x , y, z). Now since loops by definition have an identity element and [3] has shown that Moufang loops have the inverse property, if (x · y) · z = x · (y · z) for all x , y, z ∈ L, the Moufang loop L , · would be a group. It has also been observed that fixing the order of a Moufang loop can force it to be an associative Moufang loop. The aforementioned led to the study of the question: “Given a positive integer n, can we find a nonassociative Moufang loop of order n?” This is done by using the prime decomposition theorem to write n as a product of positive powers of distinct primes. These primes are usually arrayed from the smallest to the largest. The progressive and extensive works of Chein, Leong and Rajah solved completely even order Moufang loops. The odd case is still being resolved. The well-known results found so far are those of [3–7], [10–17] and [22–25]. For distinct odd primes p and q, the existence of nonassociative Moufang loops of order pq3 have been guaranteed in [22] if and only if q ≡ 1(mod p). In addition, the existence of nonassociative Moufang loops of order 34 [3] and p5 for primes p > 3 [26] have been long established. In fact, these were classified in [18] and [19], whereas the nonassociative Moufang loops of order pq3 were classified in [4]. Our work is an extension of the result in [25] where it was proved that for odd primes p and q with p < q, all Moufang loops of order p3q3 are associative if and only if q ≡ 1(mod p). Since all Moufang loops of order p4 are associative if and only if p is a prime with p > 3 [10], we move on to the next step, i.e., we obtain the following result: If p and q are primes with 3 < p < q, then all Moufang loops of order p4q3 are associative if and only if q ≡ 1(mod p). 2 Definitions and Notations The following definitions are quite standard. One can refer to [2,3,8] and [10] for further details. 1. A loop L , · , is a binary system that satisfies the following two conditions: (i) specification of any two of the elements x , y, z in the equation x · y = z uniquely determines the third element and (ii) the binary system contains an identity element (we denote it as 1). 2. A Moufang loop is a loop L , · such that (x · y) · (z · x ) = (x · (y · z)) · x for any x , y, z ∈ L. (From now on, for the sake of brevity, we shall simply refer to the loop L , · as the loop L. Also, we shall write (x · y) · z simply as x y · z, (x · (y · z)) · x as (x · yz)x , etc.) 3. The associator subloop of L is denoted as La = (L , L , L) = (l1, l2, l3)|li ∈ L . In a Moufang Loop, La is the subloop generated by all the associators (x , y, z) ∈ L such that (x , y, z) = (x · yz)−1(x y · z). It is obvious that L is associative if and only if La = {1}. 4. I (L) = R(x , y), L(x , y), T (x )|x , y ∈ L is called the inner mapping group of L, where z R(x , y) = (zx · y)(x y)−1, z L(x , y) = (yx )−1(y · x z), zT (x ) = x −1 · zx . 5. The commutator subloop of L, denoted Lc, is the subloop generated by all com mutators [x , y] in L, where x y = yx · [x , y]. 6. The subloop generated by all n ∈ L such that (n, x , y) = (x , n, y) = (x , y, n) = 1 for any x , y ∈ L is called the nucleus of L. It is denoted as N (L) or simply as N . 7. Suppose H is a subloop of L. Then CL (H ) = {g ∈ L|gh = hg for all h ∈ H }. 8. Let M be a subloop of L and π a set of primes. (i) M is a normal subloop of L, denoted M L, if M θ = M for all θ ∈ I (L). (ii) A positive integer n is a π -number if every prime divisor of n lies in π . (iii) For each positive integer n, we let nπ be the largest π -number that divides n. (iv) M is a π -loop if the order of every element of M is a π -number. (v) M is a Hall π -subloop of L if |M | = |L|π . (vi) M is a Sylow p-subloop of L if M is a Hall π -subloop of L and π contains only a single prime p. 9. Assume M is a normal subloop of L. (i) M is a proper normal subloop of L if M = L. (ii) L/M is a proper quotient loop of L if M = {1}. 10. Assume M is a normal subloop of L. (i) M is a minimal normal subloop of L if M is nontrivial and contains no proper nontrivial subloop which is normal in L. In other words, if there exists H L with {1} < H < M , then H = {1} or M . (ii) M is a maximal normal subloop of L if M is not a proper subloop of every other proper normal subloop of L. In other words, if there exists H L such that M < H , then M = H or H = L. 11. If m and n are integers, then (m, n) denotes the greatest common divisor of the two integers. 3 Basic Properties and Known Results Let L be a Moufang loop. Lemma 3.1 L is diassociative, that is, x , y is a group for any x , y in L. Moreover, if (x , y, z) = 1 for some x , y, z in L, then x , y, z is a group. [3, Moufang’s Theorem, p.117] Lemma 3.2 N = N (L) is a normal subloop of L [3, Theorem 2.1, p.114]. Clearly N , is a group by its definition. Lemma 3.3 Suppose K (a) If L/K is a group, then La ⊂ K . (b) If L/K is commutative, then Lc ⊂ K . [15, Lemma 1, p.563] Note that the properties above hold for all Moufang loops in general. However, the following properties hold only for finite Moufang loops L. (a) Then L is solvable. [8, Theorem 16, p.413] (b) If K is a minimal normal subloop of L, then K is an elementary abelian group and (K , K , L) = (k1, k2, l)|ki ∈ K , l ∈ L = {1}. [8, Theorem 7, p.402] (c) If K is a normal subloop of L , (K , K , L) = {1} and (|K |, |L/K |) = 1, then K ⊂ N . [8, Theorem 10, p.405] (d) Then L contains a (Hall π -)subloop of order |L|π . [8, Theorem 12, p.409] Lemma 3.6 L is a group if it has any of the following orders: (a) p, p2, p3 or pq where p and q are distinct primes. [6, Corollary 4 and Proposition 3, p.35] (b) pqr or p2q where p, q and r are distinct odd primes. [21, Theorem 3.1, p.124 and Theorem 3.3, p.126]. (c) p4 where p is a prime and p > 3. [10, Theorem, p.33] (d) pq2 where p and q are distinct odd primes. [13, Theorem, p.269] (e) p1 p2 · · · pm q3r1r2 · · · rn where p1, p2, · · · , pm , q, r1, r2, · · · , rn are odd primes with p1 < p2 < · · · < pm < q < r1 < r2 < · · · < rn and q ≡ 1(mod pi ) for all i ∈ {1, 2, · · · m}. [23, Theorem 4.1, p.374] (f) pαq1 · · · qn, where α ≤ 3 and p, q1, · · · , qn , are distinct odd primes with p < qi . [12, Lemma 1,2, p.349, Theorem, p.350] (g) pαq1 · · · qn, where α ≤ 4 and p, q1, · · · , qn , are distinct primes with 3 < p < qi . [15, Theorem, p.567] (h) p1α1 p2α2 · · · pnαn , where 1 ≤ αi ≤ 2 and p1, p2, · · · , pn are distinct odd primes. [14, Theorem, p.882] (i) pαq1β1 q2β2 · · · qnβn , where p and qi are primes with p < q1 < . . . < qn and βi ≤ 2 with α ≤ 3 when p > 2, or α ≤ 4 when p > 3. [16, Theorem 1, p.482 and Theorem 2, p.483] 1 2 · · · pnαn q3, where p1, p2, · · · , pn and q are distinct odd primes with q ≡ (j) pα1 pα2 1(mod pi ) and 1 ≤ αi ≤ 2. [24, Theorem 4.2, p.970] (k) p3q3, where p and q are odd primes with p < q, and q ≡ 1(mod p). [25, Theorem 4.6, p.1364] (l) pq4, where p and q are odd primes with p < q, and q ≡ 1(mod p). [5, Corollary 4.2, p.434] Lemma 3.7 Suppose |L| is odd and every proper subloop of L is a group. If there exists a minimal normal Sylow subloop in L, then L is a group. [13, Lemma 2, p.268] Lemma 3.8 If there exist H, K in L such that H then H L. [14, Lemma 1, p.879] L and (|H |, |K /H |) = 1, Lemma 3.9 Let |L| = pα1 p2α2 · · · pnαn qβ , where 1 ≤ β ≤ 2 and p1, p2, · · · , pn are 1 distinct odd primes such that pi < q. Suppose: (i) every proper subloop of L is a group, and (ii) there exists a Sylow q-subloop normal in L. Then L is a group. [14, Lemma 3, p.879] Lemma 3.10 Let L be of odd order, K , a minimal normal subloop of L such that K ⊂ L, and Q, a Hall subloop of L. Suppose all proper subloops and proper quotient loops of L are groups, (|K |, |Q|) = 1 and Q K Q. Then L is a group. [15, Lemma 3, p.564] Lemma 3.11 Let L be of odd order such that every proper subloop and proper quotient loop of L is a group. Suppose Q is a Hall subloop of L such that (|La |, |Q|) = 1 and Q La Q. Then L is a group. [15, Lemma 3, p.564] Lemma 3.12 Let L be nonassociative and of odd order such that all proper quotient loops of L are groups. Then: (a) La is a minimal normal subloop of L [23, Lemma 1(a), p.478]; and is an elementary abelian group. [8, Theorem 7, p.402] (b) if M is a maximal normal subloop of L, then La and Lc lie in M . Moreover, L = M x for any x ∈ L\M . [23, Lemma 1(b), p.478] Lemma 3.13 Suppose K is a subloop of CL (La ) and (|K |, |La |) = 1. Then K ⊂ N . [16, Lemma 5, p.480] Lemma 3.14 Suppose (a) |L| = pαm where p is a prime, ( p, m) = ( p − 1, pαm) = 1 and L has an element of order pα. Then there exists a (Sylow p-)subloop P of order pα and a normal subloop M of order m in L such that L = P M . (b) |L| = p2m where p is the smallest prime dividing |L| and ( p, m) = 1. Then there exists a subloop P of order p2 and a normal subloop M of order m in L such that L = P M . [17, Theorem 1, p.39] Lemma 3.15 Let L be of odd order and K a normal subloop of L. Suppose K ⊂ N . Then there exists a homomorphism from L to Aut (K ) with CL (K ) as the kernel. Thus, CL (K ) L and |L/CL (K )| divides | Aut (K )|. [11, Theorem 3(a), p.33] Lemma 3.16 Let L be of odd order and K a normal Hall subloop of L. Suppose K = x La for some x ∈ K \La and La ⊂ N . Then K ⊂ N . [23, Lemma 3, p.17] Lemma 3.17 Let L be nonassociative and of odd order, and let M be a maximal normal subloop of L. Suppose all proper subloops and proper quotient loops of L are groups. Then (a) La is a Sylow subloop of N ⇒ La = N . [17, Lemma 6, p.480] (b) La is cyclic ⇒ La ⊂ N . [16, Lemma 1, p.480] (c) (k, w, l) = 1 for all k ∈ La , w ∈ M, l ∈ L ⇒ La ⊂ N . [16, Lemma 3, p.479] (d) (k, w, l) = 1 for some k ∈ La , w ∈ M, l ∈ L ⇒ La contains a proper nontrivial subloop which is normal in M . [23, Lemma 3, p.19] Lemma 3.18 Suppose |L| is odd and every proper subloop of L is a group. If N contains a Hall subloop of L, then L is a group. [15, Lemma 2, p.564] Lemma 3.19 Let L be of order p1α1 p2α2 · · · pnαn q, where p1, p2, · · · , pn and q are odd primes with p1 < p2 < · · · < pn < q and α1, α2, · · · , αn ∈ Z+, q ≡ 1(mod pi ) for all i . Then there exists a normal subloop of order p1α1 p2α2 · · · pnαn in L. [25, Lemma 4.1, p.1362] Lemma 3.20 Let L be nonassociative and of odd order, and let M be an associative maximal normal subloop of L and P be a subloop of L . Suppose La ⊂ N and M = La P. Then for any x ∈ L\M , there exist some r, s ∈ P such that (x , r, s) = 1. [25, Corollary 4.3, p.1362] Lemma 3.21 Let L be of odd order and M be an associative maximal normal subloop of L. Suppose La ⊂ N . Then g commutes with (x , g, h) for any x ∈ L\M and g, h ∈ M . [25, Lemma 4.4, p.1363] Lemma 3.22 Suppose p and q are distinct odd primes. Then there exists a nonassociative Moufang loop of order pq3 if and only if q ≡ 1(mod p). [22, Theorem 1, p.78 and Theorem 2, p.86] 4 Main Results Lemma 4.1 Let G be a group and r, s, t ∈ G with [r, t ] = [s, t ] = 1. Suppose r −1sr = sαt β , for some α, β ∈ Z+. Then r −nsr n = sαn t β(α0+α1+···+αn−1) for all n ∈ Z+. Proof (By induction.) Given r, s, t ∈ G with [r, t ] = [s, t ] = 1 and r −1sr = sαt β for some α, β ∈ Z+. Let P(n) be the statement r −nsr n = sαn t β(α0+α1+···+αn−1) for n ∈ Z+. = rW−h1sern =n s=α11t β,(1α)n−1 = α0. So α0 + α1 + · · · + αn−1 = α0 = 1. Then r −nsr n = sαt β . Hence P(1) is true. Assume that P(k) is true for some fixed k ∈ Z+, i.e., r −k sr k = sαk t β(α0+α1+···+αk−1). Now r −(k+1)sr (k+1) = r −1r −k sr kr since [r, t ] = 1 since [s, t ] = 1 Hence P(k + 1) is true. Then by the principle of induction, P(n) is true for all n ∈ Z+. This completes the proof of this lemma. (Note: This proof is obtained from the proof of lemma (4.5), p.1363 in [25], which proves it for Moufang loops with an associative subloop x , y, z .) Theorem 4.2 Let L be a Moufang loop of order p4q3 where p and q are primes with 3 < p < q, q ≡ 1(mod p). Then L is a group. Proof (By contradiction.) Assume there exists L a nonassociative Moufang loop satisfying the conditions above. By lemma (3.5)(a), L is solvable. Also by Lagrange’s theorem, every proper subloop and proper quotient loop of L is of order pαqβ where α ≤ 4 and β ≤ 2 or pαq3, where α ≤ 3. All these proper subloops and proper quotient loops of L are associative by lemma (3.6)(h–k). By lemma (3.12), La is a minimal normal subloop of L. Then by lemma (3.5)(b), La is an elementary abelian group. So, |La | = p, p2, p3, p4, q, q2 or q3. If |La | = p4 or q3 then La would be a minimal normal Sylow subloop of L and so by lemma (3.7) L would be a group which contradicts assumption (4.1). So, |La | = p, p2, p3, q or q2. We prove this theorem by considering the following four cases: |La | = q, |La | = q2, |La | = p or p2 and |La | = p3. Case 1: |La| = q By lemma (3.5)(d), there exists P a Hall subloop of order p4 in L. Since La L , La P < L. So, |La P| = ||LLaa∩||PP|| = p4q. Since P is a Sylow p-subloop of La P, by lemma (3.19), P La P. Also, since (| P|, |La |) = ( p4, q) = 1, by lemma (3.10), L is a group. This contradicts assumption (4.1). Case 2: |La| = q2 By lemma (3.12) La exists a subloop M/La L. Therefore, |L/La | = p4q. Also by lemma (3.19), there L/La such that |M/La | = p4. So M L. Hence M is a maximal normal subloop of order p4q2 in L . Assume (k, w, l) = 1 for some fixed k, w, l ∈ L , with k ∈ La and w ∈ M. (4.4) Then by lemma (3.17)(d), La contains S a proper nontrivial subloop normal in M . Thus, |S| = q, so |M/S| = p4q. Then by lemma (3.19), there exists T /S M/S such that |T /S| = p4, so T M and |T | = p4q. Again by lemma (3.19), there exists R T such that |R| = p4, since R is normal Hall subloop in T , by lemma (3.8) R M . Again by lemma (3.8), R L. Since L/R is a group, by lemma (3.3)(a), La ⊂ R. Then by lemma (3.4), |La | is a divisor of |R| which is a contradiction since |La | = q2 and |R| = p4. Therefore, assumption (4.4) is false. Hence (k, w, l) = 1 for all k ∈ La , all w ∈ M and all l ∈ L . By lemma (3.17)(c) La ⊂ N . So by lemma (3.4), |La | is a divisor of |N |, i.e., q2 | |N |. Also by lemma (3.18), N cannot contain any Hall subloop of L. So q3 cannot divide |N |. Thus, La is a Sylow subloop of N . So by lemma (3.17)(a), La = N . Therefore, |N | = q2. Now by lemma (3.15), CL (N ) L and |L/CL (N )| divides | Aut (N )|. Suppose p||CL (N )|. By Sylow’s theorem there exists P a subloop of order p in CL (N ). So (|La |, | P|) = 1 and by lemma (3.13), P ⊂ N . This is a contradiction since p ||N |. Therefore, Assume |CL (N )| = q3. Now La is an abelian group by (4.2). Then by its definition, CL (La ) = CL (N ) contains La . So |La | = q2 is a divisor of |CL (N )| by lemma (3.4). Then by (4.5), |CL (N )| = q2 or q3. Then for any x ∈ CL (N )\La , | La , x | > |La |. But since La , x < CL (N ), CL (N ) = CL (La ) = La , x = x La . Now by lemma (3.15), CL (N ) is a normal Hall subloop of L, so by lemma (3.16), we have that CL (N ) ⊂ N , which is impossible since |CL (N )| = q3 and |N | = q2. So our assumption (4.6) is false. Therefore, |CL (N )| = q2. Then, By (4.3) and lemma (3.5)(d), there exists a Sylow p-subloop Now |L/M | = q. So L/M is a group. Hence by lemma (3.3), La ⊂ M Since La L, by (4.8) and (4.9), La P < M where |La P| = ||LLaa∩||PP|| = q2 p4 = |M |. Hence M = La P. By lemma (3.20), we get that: for any x ∈ L\M there exists some r, s ∈ P, such that (x , r, s) = 1. Now |La | = q2, i.e., La = N and by (4.2), Write t = (x , r, s). By, (4.11) for some u ∈ La \ t . So by lemma (3.21) La = Cq × Cq . La = t × u [r, t ] = 1, since P ⊂ M by (4.8). The fact that La L and u ∈ La means r −1ur ∈ La . So by (4.12) we can express for some λ, η ∈ Z+. Now (u, t, r ) = 1 since t ∈ La = N . Thus, the elements u, t and r associate. So by lemma (4.1), we get that u = r −|r|ur |r| = uλ|r| t η(λ0+λ1+···+λ|r|−1) since |r | ∈ Z+ as r = 1 by (4.10). Then u1−λ|r| = t η(λ0+λ1+···+λ|r|−1) = 1. Since t ∩ u = {1}, u1−λ|r| = 1. So q|(1 − λ|r|) ⇒ λ|r| ≡ 1(mod q). Now since r ∈ P, | P| = p4 and q ≡ 1(mod p), (|r |, q − 1) = 1. It follows that the congruence λ|r| ≡ 1(mod q) has only one solution for λ, i.e., λ = 1. Now λ0 + λ1 + · · · + λ|r|−1 = (1 + 1 + · · · + 1) = |r |. So t η(λ0+λ1+···+λ|r|−1) = |r| times t η|r| = 1. Thus, |t | = q divides η|r |. Since q is not a factor of |r |, q|η. This means t η = 1. Hence r −1ur = u by (4.14), i.e., [r, u] = 1. So, by (4.12) and (4.13), r commutes with both generators of La . Therefore, r ∈ CL (La ) = CL (N ) = N by (4.7). Then (x , r, s) = 1 by the definition of N . This is a contradiction since (x , r, s) = 1 by (4.10). Therefore, |La | = q2. Case 3: |L a| = pµ, 1 ≤ µ ≤ 2. |L a Q| = ||LLaa ∩||QQ|| = pμq3. Since Q is a Sylow q-subloop of L a Q, by lemma (3.14)(b), Q L a Q. Since (|Q|, |L a |) = (q3, pμ) = 1, by lemma (3.10), L is a group. This again contradicts assumption (4.1). Case 4: |L a| = p3 By lemma (3.12), L a L . Since |L /L a | = pq3, by lemma (3.14), there exists a subloop M/L a L /L a such that |M/L a | = q3. So M L with |M | = p3q3. Hence M is a maximal normal subloop of L . Assume (k, w, l) = 1 for some k ∈ L a , w ∈ M and l ∈ L By lemma (3.17)(d), L a contains S a proper nontrivial subloop normal in M . Thus, |S| = pγ , 1 ≤ γ ≤ 2. So |M/S| = p3−γ q3, 1 ≤ 3 − γ ≤ 2 and by lemma (3.14), there exists T /S M/S such that |T /S| = q3. So |T | = pγ q3 with T M . Again by lemma (3.14), there exists R T such that | R| = q3, and since R is normal Hall subloop in T , by lemma (3.8) T M . Again, since R is normal Hall subloop in M by lemma (3.8), R L . Since L / R is a group, by lemma (3.3)(a) L a ⊂ T . Then by lemma (3.4), |L a | is a divisor of | R| which is a contradiction since |L a | = p3 and | R| = q3. Therefore, assumption (4.15) is false. Hence (k, w, l) = 1 for all k ∈ L a , w ∈ M and l ∈ L . By lemma (3.17)(c) L a ⊂ N . So by lemma (3.4), |L a | = p3 is a divisor of |N |. Also by lemma (3.18), N cannot contain a Hall subloop of L . So p4 cannot divide |N |. Thus, L a is a Sylow subloop of N . So by lemma (3.17)(a), L a = N . Therefore, |N | = p3. Now by lemma (3.15), the subloop CL (N ) L and |L /CL (N )| divides | Aut (N )|. Assume q||CL (N )|. Then, by Sylow’s theorem there exists Q a subloop of order q in CL (N ). So (|L a |, |Q|) = 1 and by lemma (3.13), Q ⊂ N . This is a contradiction since q ||N |. Therefore, Now L a is abelian by (4.2). Then by its definition, CL (N ) = CL (L a ) contains L a . So |L a | = p3 is a divisor of |CL (N )|. So by (4.16), |CL (N )| = p3 or p4. Assume |CL (N )| = p4. Then there exists x ∈ CL (N )\L a such that L a , x < CL (N ) with | L a , x | > |L a |, i.e., L a is a proper subloop of CL (N ). But since L a , x < CL (N ), CL (N ) = L a , x = x L a . Now by lemma (3.15), CL (N ) is a normal Hall subloop of L . So by lemma (3.16) we have that CL (N ) ⊂ N , which is impossible since |CL (N )| = p4 and |N | = p3. So |CL (N )| = p4. Therefore, |CL (N )| = p3, i.e., CL (N ) = CL (La ) = La = N . So |L/CL (La )| = pq3. Then by lemma (3.15), Since |La | = p3 in this case (i.e., case 4), La = C p × C p × C p by (4.2). Then by [1], Aut (La ) ∼= G L(3, p) by simply viewing La as a vector space of dimension 3 over C p, where |G L(3, p)| = ( p3 − 1)( p3 − p)( p3 − p2). Therefore, | Aut (La )| = ( p3 − 1)( p3 − p)( p3 − p2) = p3( p3 − 1)( p2 − 1)( p − 1) = p3( p − 1)3( p + 1)( p2 + p + 1). Then by (4.17), q3| p2( p − 1)3( p + 1)( p2 + p + 1). Now p − 1 < p < q ⇒ q | p and q |(pp2(−p 1−).1A)3ls(opq+≡1)1.(mod p) ⇒ q |( p + 1). Since q is a prime, q | Therefore, q3|( p2 + p + 1) and so q3 ≤ p2 + p + 1 < p2 + 2 p + 1 = ( p + 1)2. Thus, However, since both p and q are odd primes, with p < q, p + 2 ≤ q ⇒ p + 1 < q. So ( p + 1)2 < q2. Then we have by (4.18) that q3 < q2 which is a contradiction. Therefore, |La | = p3. Since each case 1, 2, 3 and 4 ended with a contradiction, we conclude that assumption (4.1) is false. Hence L is a group. 5 Open Problem In view of lemma (3.6) (l) and theorem (4.2), the next unsolved case is Moufang loops of order p2q4 where p and q are odd primes with p < q and q ≡ 1(mod p). Recently we have obtained a partial result for this problem by adding a sufficient—but perhaps unnecessary—condition q ≡ −1(mod p) and proving that all such Moufang loops are associative. Due to the difficulty we face in proving associativity without this added condition, we are making a bold (but perhaps untrue) conjecture: “There exist (a class of) nonassociative Moufang loops of order p2q4 where p and q are odd primes with p < q, q ≡ 1(mod p) but q ≡ −1(mod p).” If our conjecture is true, there would exist a nonassociative Moufang loop of order 3254—the smallest one satisfying the given conditions. 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Andrew Rajah, Lois Adewoye Ademola. Moufang Loops of Odd order \(p^4q^3\), Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1-12, DOI: 10.1007/s40840-017-0471-2