Generalizing Camina groups and their character tables

Abstract

Abstract. We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are shared by these generalized Camina groups. In particular, we show that if G is a nilpotent, generalized Camina group then its nilpotence class is at most 3. We use the information we collect about generalized Camina groups with nilpotence class 3 to characterize the character tables of these groups.

The vanishing-off subgroup

“…We should note that there are some important variances in this result from the results of [11]. In particular, we were able to prove that |G : C | = |G : V 2 (G)|(= |G : G 3 |) is a square whenever G is a generalized Camina group of nilpotence class 3.…”

“…This additional assumption is actually necessary, as we have examples where |G : C | is not a square. Following [11], it is tempting to conjecture that |G 3 : V 3 (G)| |G : V 2 (G)|, however, this need not be true since we have an example where |G 3 : V 3 (G)| = |G : V 1 (G)| = 4 and |G : V 2 (G)| = 2. In any case, we would like to obtain some bound for |G i : V i (G)| in terms of |G : V (G)| when i 3.…”

“…We obtain information regarding the terms of this series. In particular, we will show that all of the quotients G i /V i (G) will be elementary abelian p-groups for some prime p. In addition, we will show that G/V 2 (G) will be what we called a VZ-group in [10] and [11] and those are the class of groups studied extensively in [5].…”

“…In [11], we studied many of the properties of generalized Camina groups. We will show that many of the properties of generalized Camina groups can be found in general (nilpotent) groups using the vanishing-off subgroup.…”

Relative Elementary Abelian Groups and a Class of Edge-Transitive Cayley Graphs

“…A group G is called a Camina group if all elements of gG with g G are conjugate to g (refer to [2,3,10,11,13]). The concept of REA group is in some sense a generalisation of Camina group.…”