On the prime graph of the unit group of integral group rings of finite groups

Integral Group Ring of the Mathieu Simple GroupM23

“…The Luthar-Passi method proved to be useful for groups containing non-trivial normal subgroups as well. Also some related properties and some weakened variations of the Zassenhaus conjecture as well can be found in [1,22] and [3,20]. For some recent results we refer to [5,7,15,16,17,18].…”

“…By #(G) we denote the set of all primes dividing the order of G. The Gruenberg-Kegel graph (or the prime graph) of G is the graph π(G) with vertices labeled by the primes in #(G) and with an edge from p to q if there is an element of order pq in the group G. The following weakened variation of the Zassenhaus conjecture was proposed in [20]:…”

“…Throughout the paper we will use the notation, inclusive the indexation, for the characters and conjugacy classes as used in the GAP Character Table Library. Since the group G possesses elements of orders 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 21 and 23, first of all we will investigate units of some of these orders (except units of orders 4, 6, 8, 12 and 14). After this, by Proposition 4, the order of each torsion unit divides the exponent of G, and in the first instance we should consider units of orders 20,22,24,28,30,33,35,42,46, 55, 69, 77, 115, 161 and 253. We will omit orders 20, 24, 28, 30 and 42 that do not contribute to (KC), and this enforces us to add to the list of exceptions in part (i) of Theorem also orders 40, 56, 60, 84, 120 and 168, but no more because of restrictions imposed by the exponent of G.…”

Torsion units for a Ree group, Tits group and a Steinberg triality group

“…This question is upheld for Frobenius and Solvable groups in [20]. It was also confirmed for some Sporadic Simple groups in [21][22][23][24][25][26][27][28][29][30][31][32].…”

Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order

“…A significant sketch of the proof that the complex group algebra of a finite simple group G determines G up to isomorphism provided G is a sporadic simple group or G is isomorphic to an alternating group A n has been given in [12]. For the case of alternating groups a complete proof has been worked out in the author's Diplomarbeit [15].…”