# Algebra Carbondale Nineteen Eighty: Proceedings by Amayo R.K. (ed.) By Amayo R.K. (ed.)

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Bbk, bbk) (21 +1)2+212 (21 + 1). Therefore (6-b, bkTk) where ai C= B = Z= more 1) (21 + hand, other = 1). g. we assume may ,3plapl (21 6japl Since = 1 and In T,,,,(21 = x, y, xx will admit Yx V-y = fxy) = out certain those 1, we (21 for all 1). + 1) (1 + = On the other 1) bk+1 that which 1). + 21(l > obtain + T. Then Bb 0 -< i < m. 011apl. = = G a, is a Nowthe hand Therefore B. linear span of isomorphism 0 immediately. 4) nil. [nl+l, that such that elements basis = Some small available. Remark 2.

27) reads and = 7 + = , 4(b3,b2 2) =3+3(b3,bb3)+(b3,b3F3)51 + 2b2 + b3 + 73, (T3, bb3) 0 and b3T3 (b3, b2)2 =,4 0, so that b22 51+b+b+b3 +b3. , 45 a = (51 + b + (Tb3, bb3) = (N + Hence p 20. = (bb3 bb3) = (6-b, b373) = , 2b2 + q = follows T + b3 Element Degrpe of -1 ((b3 U) 2 2 7 5 69 b3) ((b3 1) - that + F3, T3- p, b + b3) 3b + b + + b3 q) + 25 5 + = (p, q), 2b5 and = (51 = 1 = 7 It (b 3F37 b 2) Nonreal (b3 W) = = - Faithful a T + 2b2, + b + T + b3 51 + b + + 73) 25, contradiction. , 2 (b2 , 7 In u sequel the follows it case = T (b, b3T3) = (b7 b3T3) = = 7 51 + b + = becomes I and contradicting that 2 b3 reads b3 is real.

3plapl 1) 21(l > have the property: are -2; = determine not Pplapl 21 + 1 is odd and whenever turn a + + + implies index such that b, will we Then Epi-1,31lail 1)al + follows which then It . , B,,, 1bk = m be a Let by 21 is divisible bbk Op. : 1)(1 + 1)2+ (21 +1)12. 4). 4) imply' already values Lemma 5). (cf. 4) holds. It is Even holds property problem counterexamples in are provided an open to are Z. Arad et al. 48 Proof. already If x, y, (a) 2 + If b2 (i) (ii) (iii) (b) [xyl 1). :7 T3, 73, b2T2 = (I First the 1 for = 1 + 1 for = note prime proofs (a) By n = 21 + 1 is = > = of b272 or Icl jb2j 2(n is multiple a 1) - (a) (ii) equivalent: are equivalent: (n2 +I)n 0 (b2 b3) b2 b3) b4), b3 0 b4; + of -==> E a c for n all which (bA, b2b3) < is 1(b3 (c, b2b3) (iii) =, are 2 calculation.