Let N be a normal subgroup of a **p**-solvable group **G**. The purpose of this paper is to investigate some properties of N under the condition that the two longest **sizes** of the non-central **p**-**regular** **G**-**conjugacy** classes of N are coprime. Some known results are generalized.

Lemma . Let **G** be a simple group with a disconnected prime graph. Then its order com- ponents are exhibited in Tables -, where **p** is an odd prime and q is a prime power. Proof By [, , ] and the deﬁnition of order component, we easily get order components of **G** in Tables -. Note that some mistakes and misprints in [, , ] are amended in

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In Chapter 2, literature on related work is perused. The focus being on the cycle index formulas, subdegrees and suborbital graphs of permutation groups. In Chapter 3, intersections of **conjugacy** **class** of elements **g** ∈ **G** with certain subgroups of **G**, are used to determine the disjoint cycle structure of **g** and hence cycle index of **G** corresponding to some given permutation representations. In essence, the action of **P** SL(2, q) on the cosets of C q−1

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Let **G** be a group with identity element e and x, y ∈ **G**. The element x is conjugate to y if there exists an element **g** ∈ **G** such that **g** −1 xg = x **g** = y. The relation x is conjugate to y is an equivalence relation on **G**. It is reflexive since x e = x and symmetric since if x **g** = y then x = gyg −1 = y **g** −1 . Finally it is transitive since if x **g** = y and y h = z then substitute x **g** for y, we get (x **g** ) h = z. Set

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Abstract. We determine the maximal number of **conjugacy** classes of maximal elementary abelian subgroups of rank 2 in a finite **p**-group **G**, for an odd prime **p**. Namely, it is **p** if **G** has rank at least 3 and it is **p** + 1 if **G** has rank 2. More precisely, if **G** has rank 2, there are exactly 1, 2, **p** + 1, or possibly 3 classes for some 3-groups of maximal nilpotency **class**.

Let **G** be a group with the identity element e and x, y ∈ **G**. The element x is conjugate to y in **G** if there exists an element **g** ∈ **G** such that **g** −1 xg = x **g** = y. The relation x is conjugate to y is an equivalence relation on **G**. This equivalence relation induces a partition of **G** whose elements are called **conjugacy** classes. The number of **conjugacy** classes of **G** is denoted by cl **G** .

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Theorem 2.14. If S is one of the groups in Table 1 (up to isomorphism), then **G** ∼ = S × Z(**G**). Proof. Since S is OC-characterizable, by the references stated in the third column of Table 1, we deduce from Theorem 2.13 that if M(S) = 1, then **G** ∼ = S × Z (**G**), as desired. In the following, we are going to study the remaining cases, with the help of [1]. For this aim let H be a covering group of S such that S ̸∼ = H. Then considering M(S) shows that:

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Comments. All results of this section are from [41]. It is still un- known: does there exist in ZFC a nondiscrete nodec topological group or a nondiscrete extremally disconnected topological group? Under MA, ev- ery countable Abelian group admits a nondiscrete nodec group topology (see [58] or [35, Chapter 2]). In the case of countable groups, the sec- ond question is an old problem of Arhangel’ski ˘ i [2]. Under CH, the first example of nondiscrete countable extremally disconnected group was con- structed in [51]. Theorem 10.10 is a generalization of the Sirota’s results. Some partial results incline us to the negative answer to the Arhangel’ski ˘ i question. Thus, Theorem 9.6 can be re-formulated as follows. If there exists a countable nondiscrete extremally disconnected topological group **G** such that the topology of **G** is maximal in the **class** of all **regular** left invariant topologies on **G**, then there is a **P** -point in ω ∗ . Another

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with another random element from the same group. In this paper we found the probability in the case n = 1 where **G** is a 2-generator 2-groups of nilpotency **class** 2 up to order 64, using the fact that the probability is the number of **conjugacy** classes divided by the order of the group. We also computed and presented the probability **P** n (**G**) for n in general ( ne).

be noted that similar results have also been studied for other graphs. In [19], S. Dolfi defines the dual graph of Γ(**G**) whose vertices are the primes which occur as divisors of the **class** **sizes** of **G**, and two vertices **p** and q are joined by an edge if there exists a **conjugacy** **class** in **G** whose size is a multiple of pq. In [16], Dolfi and C. Casolo describe all finite groups **G** for which the dual graph of Γ(**G**) is connected and has diameter three. The corresponding dual graph of Γ **G** (N ) was defined in

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for each **G**; these are Tables 1A–5A. Each row of the table gives a **conjugacy** **class** of **G**-irreducible connected subgroups, with the first column giving the ID number, the second column the isomorphism type of a representative X and the third column gives any restrictions on the characteristic **p**. The fourth column gives all **conjugacy** classes of immediate connected overgroups of X . Note that we use the notation X [#n] to denote the subgroup X = **G**(#n), as a shorthand only in these five tables. If X is a diagonal subgroup of Y and all immediate connected overgroups of X are also subgroups of Y (and hence diagonal subgroups of Y or just Y itself) then X does not appear in the auxiliary table. In this case there will be only one appearance of X in Tables 1–5 (or the supplementary tables in Section 11.1) and all of its immediate overgroups can be straightforwardly computed from the other diagonal subgroups that will appear just above X in the table. For this reason, the phrase “immediate overgroups in Y [#n]” sometimes appears in the fourth column of a row corresponding to the irreducible subgroup X . This means that the subgroup X is a diagonal subgroup of Y and the immediate connected overgroups of X in Y are included in the list but not explicitly calculated.

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The notion of the bipartite divisor graph was first introduced by Iranmanesh and Praeger in [4] for a finite set of positive integers. As an application of this graph in group theory, in [2], the writers considered this graph for the set of **conjugacy** **class** **sizes** of a finite group and studied various properties of it. In particular they proved that the diameter of this graph is at most six, and classified those groups for which the bipartite divisor graphs of **conjugacy** **class** **sizes** have diameter exactly 6. Moreover, they showed that if the graph is acyclic, then the diameter is at most five and they classified the groups for which the graph is a path of length five. Similarly, Taeri in [20] considered the case that the bipartite divisor graph of the set of **conjugacy** **class** **sizes** is a cycle and (by using the structure of F -groups and the classification of finite simple groups) proved that for a finite nonabelian group **G**, the bipartite divisor graph of the **conjugacy** **class** **sizes** is a cycle if and only if it is a cycle of length 6, and for an abelian group A and q ∈ { 4, 8 } , **G** ≃ A × SL 2 (q). Inspired by these papers, in this work we consider

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Proof. In light of Lemmas 28–30 we can assume that q ≥ 11. Define q + = q+1 k , where k = gcd(2, q + 1), and similarly for q − . Jones proves in [29] that hyperbolic generating triples of type (**p**, q − , q − ) exist for **G** when q ≥ 11 and since gcd(**p**, q − ) = 1 we immediately have, by Lemma 16, a hyperbolic generating triple for **G** × **G**. We proceed to show that there exists a hyperbolic generating triple (x, y, z) for **G** of type (q + , q + , q + ) and note that both **p** and q − are coprime to q + . The only maximal subgroups containing elements of order q + are the dihedral groups of order 2q + which we denote by D q + . By Gow’s Theorem, for a **conjugacy** **class**, C, of

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As Taeri comments, B(**G**) having no cycle of length 4 is equivalent to **G** satisfying the one-prime power hypothesis, that is, if m and n are distinct non-trivial **conjugacy** **class** **sizes** of **G** then either m and n are coprime or their greatest common divisor is a prime power. This is similar to the one-prime hypothesis introduced by Lewis to study character degrees [11]. We use this terminology.

First, in Section 2, we pay attention to **conjugacy** **class** **sizes** of some elements of a group not divisible by **p** 2 , for some fixed prime **p**. Next, we consider square-free **class** **sizes** for all primes. Finally, in Section 4, the topic of **conjugacy** **class** **sizes** is combined with factorised groups, and some interesting achievements are shown.

In the past years, the probabilistic theory was proved to be beneficial in the solution of different complicated problems in group theory. The question that come arised from the previous researches was “Can someone determine the abelianness for a nonabelian group?” Let **G** be a group and let x, y be elements in **G**. We consider the total number of pairs of x y , for which x and y commute i.e xy yx and then divide it by the total number of pairs of x y , which is possible. Thus, the result will give the probability that a random pair of element in a group commutes or the commutativity degree of a group **G**.

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there is at least one (usually many) elements of maximal length and excess zero in every **conjugacy** **class**. Finally it is easy to check that every element of the dihe- dral group has excess zero, so the result is trivially true. Thus Theorem 1.1 holds for every finite irreducible Coxeter group, and hence for all finite Coxeter groups. Theorem 1.1 shows the existence of at least one element of maximal length and excess zero in every **conjugacy** **class** of a finite Coxeter group. However, if one looks at some small examples in the classical Weyl groups, it appears that every element of maximal length in a **conjugacy** **class** has excess zero. It is natural to ask whether this holds in general. It turns out that it does not – although the number of elements for which it fails seems to be small. For example, if W is of type E 6 , then in 23 of

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centric connectivity index of Γ G for every non-abelian finite group G in terms of the number of conjugacy classes k ( G ) and the size of the group G.. Keywords: non-commuting graph, ec[r]

In this paper we study groups whose co-word problem with respect to some ﬁnite generating set (and therefore, it turns out, with respect to any ﬁnite generating set) is a context-free language. For brevity we call such groups co-context-free (co CF) groups. Notice that, since the **class** of **regular** languages is closed under complementation, groups with **regular** co-word problem are precisely the ones with **regular** word problem, or equivalently all ﬁnite groups [1].

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In December 2009, following a change in Cabinet Secretary from Fiona Hyslop to Michael Russell, the Government proposed a Framework Agreement on **class** **sizes**, free school meals and pre-school nursery hours (Scottish Government online). While the long term aim of lower **class** **sizes** remained, this agreement set out a target for progress by August 2010. This was that 20% of pupils in P1-3 would be in classes of 18 by August 2010. In return, local authorities were given ‗flexibility‘ in relation to other Concordat commitments. The increase in pre-school hours from 475 to 570 a year would not be set in legislation, although councils were encouraged to work towards it. The original concordat commitment for free school meals for all P1-3 pupils by August 2010 was altered to a free school meal (either breakfast or lunch) in P1-3 in the 20% most deprived communities. However, the agreement re-iterated the long term aspiration:

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