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Solvable Groups Whose Character Degree Graphs Generalize Squares J. Group Theory 23 (2020), 217–234 DOI 10.1515/jgth-2019-0027 © de Gruyter 2020 Solvable groups whose character degree graphs generalize squares Mark L. Lewis and Qingyun Meng Communicated by Robert M. Guralnick Abstract. Let G be a solvable group, and let .G/ be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that .G/ is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and .G/ is block square, then G is a direct product of subgroups having disconnected character degree graphs. 1 Introduction Throughout this paper, G will be a finite group, and cd.G/ .1/ Irr.G/ D ¹ j 2 º will be the set of irreducible character degrees of G. The idea of studying the in- fluence of cd.G/ on the structure of G via graphs related to cd.G/ was introduced in [9]. The prime divisor character degree graph which is denoted by .G/ for G. We will often shorten the name of this graph to the character degree graph. The vertex set for .G/ is .G/, the set of all the prime divisors of cd.G/, and there is an edge between two vertices p and q if pq divides some degree a cd.G/. 2 The character degree graph has proven to be a useful tool to study the normal structure of finite solvable groups when given information about cd.G/. For exam- ple, the first author classifies the solvable groups whose character degree graphs are disconnected in [4]. We say that the group G is disconnected when .G/ is disconnected. In both [2] and [13], structural information is obtained for solv- able groups whose character degree graphs have diameter 3. In [10], C. P. Morresi Zuccari shows if G is a solvable group whose Fitting subgroup is abelian and .G/ has no vertex adjacent to all of the other vertices in the graph, then G is a direct product of disconnected subgroups. In [7], we prove for a solvable group G that if .G/ is a square, then G is a direct product of two subgroups of G that each have disconnected graphs con- This work was supported by China Scholarship Council (CSC) and National Science Foundation Council (NSFC) (Grant Nos. 11601121, 11771356) and Science Foundation of Henan University of Technology (Grant No. 31490036). Corresponding author is Qingyun Meng. 218 M. L. Lewis and Q. Meng taining two vertices. A natural way to generalize this graph is to replace each vertex by a set of vertices. With this in mind, a (simple) graph is said to be a block square if its vertex set can be written as a union of four disjoint, nonempty subsets 1; 2; 3; 4, where no prime in 1 is adjacent to any prime in 3 and no prime in 2 is adjacent to any prime in 4, and there exist vertices in both 1 and 3 that are adjacent to vertices in 2 and in 4. Notice that if H and K are disconnected groups whose graphs each have two connected components and .H / .K/ ¿, then .H K/ will be a block \ D square. The question arises: are there other ways to obtain a block square graph with a solvable group, or can we generalize our result in [7] from squares to block squares. We are unable at this time to fully answer this question. However, we do obtain a partial answer. First, we see using [4] that if G is solvable and disconnected, then G has at most one nonabelian normal Sylow subgroup. Hence, if G H K, where H and K D are solvable and disconnected, then G can have at most two nonabelian normal Sylow subgroups. Hence, our first result can be considered one small indication that being block square implies the direct product property. Theorem 1.1. If G is a solvable group, where .G/ is a block square, then G has at most two normal nonabelian Sylow subgroups. Second, there are two situations where we are able to prove that G is a direct product of groups with disconnected graphs when G is solvable and .G/ is block square. The first is when G has two normal nonabelian Sylow subgroups. Theorem 1.2. Let G be a solvable group, where .G/ is a block square. If G has two normal nonabelian Sylow subgroups, then G M N , where M and N are D disconnected subgroups of G that satisfy .M / .N / ¿. \ D The second is the case where G has one normal Sylow subgroup and the block containing the prime for this normal subgroup and the adjacent block are all single points. Theorem 1.3. Let G be a solvable group with a normal nonabelian Sylow p-sub- group P . Suppose .G/ is a block square whose vertex set is .G/ p;r;s , D ¹ º [ where p is adjacent to both r and s but not adjacent to any prime in and both r and s are both adjacent to at least one vertex in but r is not adjacent to s. Then G A B, where .A/ p and .B/ r; s . D D ¹ º [ D ¹ º In this paper, we study the normal structure of a finite solvable group G whose character degree graph .G/ is a block square. At this time, it is still an open question as to whether or not G must be a direct product of disconnected groups Solvable groups whose character degree graphs generalize squares 219 when G is solvable, where .G/ is block square and G has no nonabelian normal Sylow subgroups or one normal nonabelian Sylow subgroup and .G/ is not the particular structure studied in Theorem 1.3. 2 Preliminaries We start by establishing some notation which will be used repeatedly. Given an integer n, define .n/ to be the set of all the primes dividing n. When G is a group and H G, we write .G H/ for . G H /. Let N be a normal subgroup Ä W j W j of G. Use NL.N / to denote the set of nonlinear irreducible characters of N . For a character Irr.N /, we define Irr.G / for the set of irreducible constituents 2 j of G and cd.G / .1/ Irr.G / . We set j D ¹ j 2 j º 0.G/ p .G/ there exists P Syl .G/; P C G : D ¹ 2 j 2 p º For other notation on group theory, the reader is referred to [3]. We remind the reader that there have been a large number of results proved about character degree graphs. A nice expository article with references covering many of these results is [6]. One result that we need is Pálfy’s condition (see [12]) which states that if G is solvable and u; v; w are vertices in .G/, then there ¹ º must be an edge incident to two of u; v; w . One consequence of this is a theorem ¹ º of Manz and Wolf which is that if G is solvable, then .G/ has diameter at most 3. We note that Pálfy’s condition implies that if G is solvable group so that .G/ is a block square, then each i induces a complete subgraph of .G/. The next lemma is useful to evaluate the cardinality of 0.G/ in terms of the structure of .G/. It is proved as [11, Lemma 3.2], so we do not prove it here. Lemma 2.1. Let G be a solvable group. Given a prime q .G/, set ı.q/ to be 2 the set of all vertices in .G/ that are different from q and not adjacent to q in .G/. Then 0.G/ ı.q/ 1. j \ j Ä We translate this lemma to the situation where .G/ is a block square. Note that this is Theorem 1.1 in the introduction. Theorem 2.2. Let G be a solvable group, and assume that .G/ is a block square. Then i 0.G/ 1, i 1; : : : ; 4, and in particular, 0.G/ 2. j \ j Ä D j j Ä Proof. Observe for each i that there is some vertex of .G/ that is not adjacent to all of the vertices in i . Hence, we may apply Lemma 2.1 to each set i , and we have i 0.G/ 1, i 1; : : : ; 4. On the other hand, the vertices in 0.G/ j \ j Ä D induce a complete subgraph of .G/. Observe that a complete subgraph of .G/ can contain vertices from at most two i . Thus, there are at most two i having vertices in 0.G/, and the result follows. 220 M. L. Lewis and Q. Meng Sometimes, when dealing with subgroups and quotients of a solvable group whose character degree graph is block square, one will encounter solvable groups whose character degree graphs are disconnected, 1-connected, or have diameter 3. We will use the following properties of G when .G/ has one of these structures. The first result is proved as [4, Main Theorem and Lemmas 3.1–3.6]. Lemma 2.3. If G is a solvable group so that .G/ is disconnected, then the fol- lowing are true. (a) G satisfies the hypotheses of one of [4, Examples 2.1–2.6, Section 2].
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