By W. B. Vasantha Kandasamy
For the 1st time, each finite crew is represented within the kind of a graph during this e-book. This research is critical simply because houses of teams could be instantly acquired via taking a look at the graphs of the teams.
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Example text
Thus D28 = {1, a, b, b2, b3, b4, b5, b6, b7, ab, ab2, ab3, ab4, ab5, ab6, ab7} and its identity graph is as follows. 6 which can contribute to S are as follows. H1 = {1, b, b2, b3, b4, b5, b6, b7} H2 = {1, a}, H3 = {1, ab}, H4 = {1, ab2}, H5 = {1, ab3}, H6 = {1, ab4}, H7 = {1, ab5}, H8 = {1, ab6} and H9 = {1, ab7}. 8 = *H i . Now F (S) = 3. 47: Let ° §1 2 3 4 · A4 = ®e ¨ ¸ , p1 ¯° ©1 2 3 4 ¹ § 1 2 3 4· ¨ ¸, © 2 1 4 3¹ p2 §1 2 3 4· ¨ ¸ , p3 ©3 4 1 2¹ §1 2 3 4· ¨ ¸, ©4 3 2 1¹ p4 §1 2 3 4 · ¨ ¸ , p5 ©1 3 4 2 ¹ §1 2 3 4 · ¨ ¸, ©1 4 2 3 ¹ p6 §1 2 3 4· ¨ ¸ , p7 ©3 2 4 1¹ § 1 2 3 4· ¨ ¸, © 4 2 1 3¹ p8 §1 2 3 4· ¨ ¸ , p9 ©2 4 3 1¹ §1 2 3 4· ¨ ¸, © 4 1 3 2¹ p10 § 1 2 3 4· ¨ ¸ , p11 © 2 3 1 4¹ § 1 2 3 4 · °½ ¨ ¸¾ © 3 1 2 4 ¹ °¿ be the alternating group of S4.
G has one and only one subgroup. , G is a single colourable bad group. Now we give a theorem which guarantees the existence of single colourable bad groups. 7: Let G = g | g p 2 1 where p is a prime, be a cyclic group of order p2. G is a single colourable bad group. Proof: Let G = g | g p 2 1 = {1, g, g2, … g p 2 1 } be a cyclic 2 group of order p , p a prime. The only subgroup of G is H = {gp, g2p, g3p, …, g(p–1)p, 1}. Clearly order of H is p. Further G has no other subgroups. The special graph of G is formed by (p2 – 1)/ 2 triangles centered around 1.
31: Let G = ¢g | g13 = 1² be the cyclic group of order 13. 34 This has no special identity subgraphs. 32: Let G = ¢g | g16 = 1² be the cyclic group of order 16. 35 The subgroups of G are H1 = {g2, g4, g6, g8, g10, g12, g14, 1}, H2 = {g4, g8, g12, 1} and H3 = {1, g8}. 33: Let G = ¢g | g9 = 1² be the cyclic group of order 9. 36 38 The subgroup of G is H = {1, g3, g6}. 3: If G = ¢g | gn = 1² be a cyclic group of order n, n an odd number then the identity graph Gi of G is formed with (n – 1) / 2 triangles.