On Groups Whose Orders are Products of Three Prime Factors

Cole, F. N.; Glover, Jr

doi:10.2307/2369839

Cited by 10 publications

(6 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When |G| is divisible by multiple primes, the Sylow Theorems allow us to explore the p-group components for each prime p in the prime factorization of |G|, however it is unclear exactly how those components will interact with one another. In the nicest situation, when G is nilpotent, then G can be written as a direct product of its Sylow subgroups 4 and we may apply Theorem 1.1 to directly count the number of subgroups. Slightly more generally, whenever G can be expressed as a direct product of subgroups with coprime orders then this avenue will be available to us -i.e.…”

Section: Non-abelian Groups With |G| Divisible By Multiple Primesmentioning

confidence: 99%

See 1 more Smart Citation

Classifying Groups With a Small Number of Subgroups

Betz

Nash²

2022

The American Mathematical Monthly

View full text Add to dashboard Cite

We provide lower bounds on the number of subgroups of a group G as a function of the primes and exponents appearing in the prime factorization of |G|. Using these bounds, we classify all abelian groups with 22 or fewer subgroups, and all non-abelian groups with 19 or fewer subgroups. This allows us to extend the integer sequence A274847 [13] introduced by Slattery in [12].It is a classic problem in a first course in group theory to show that a group G has exactly two subgroups if and only if G ∼ = Z p for a prime p. The main idea here is to observe that if | Sub G| = 2 or if G ∼ = Z p , then every non-identity element x ∈ G must by necessity generate all of G, i.e. x = G for all x ∈ G. Slightly less frequently, a course may follow up by considering groups G with exactly three or four subgroups. In those cases, it turns out that we can again argue that G must be cyclic.Indeed, if | Sub G| = 3 and H ≤ G is the unique, non-trivial, proper subgroup of G, then observe that for any x ∈ G \ H, we must have x = G as these elements are non-trivial, must generate a subgroup of G, and cannot generate the trivial subgroup or H. Since G must be cyclic, the fact that cyclic groups have exactly one subgroup for each positive divisor of |G| implies that |G| = p 2 for some prime p and thus G ∼ = Z p 2 as cyclic groups of order |G| are unique up to isomorphism.Similarly, if | Sub G| = 4 and H = K are the two non-trivial subgroups, then recall thatOnce again, x = G and G is cyclic. As before, a cyclic group G must have exactly one subgroup for each divisor of |G|, hence it follows that G ∼ = Z pq or Z p 3 for primes p and q. We summarize these classic results below. Classic Results.1. If | Sub G| = 2, then G ∼ = Z p for some prime p. Iffor some primes p and q.These classic results beg the question: Which (necessarily finite) groups G have exactly k subgroups when k ≥ 5? Fortunately this becomes much more interesting moving forward as G need not be cyclic when | Sub G| ≥ 5. Thus, from here on we will need a completely different approach.Miller explored this topic previously in a series of obscure and terse papers [6,7,8,9,10] in which he claims to classify the groups with 16 or fewer subgroups, but it is unclear to the authors exactly how he arrives at his conclusions. Despite that, his results agree with ours, except in the case when | Sub G| = 14 where he seems to have skipped a case, causing him to miss S 3 × Z 3 and Z 3 ⋊ Z 32 . Given the assertions within, it is certainly clear that Miller is not applying the techniques we use here. Recently, Slattery [12] explored this idea once more; reducing any group G by factoring out any cyclic central Sylow p-subgroups of G first. Using this method, he worked to classify groups with 12 or fewer subgroups up to similarity defined in the following sense: Definition (From [12]). Let G and H be finite groups.

show abstract

Section: Non-abelian Groups With |G| Divisible By Multiple Primesmentioning

confidence: 99%

“…It is known (see e.g. [4]) in a group of order pqr with p < q < r, that R ∈ Syl r (G) must be a normal subgroup. Hence there exist product subgroups H pr and H qr , the latter of which must be normal as it has index p (again, see Theorem 1 in [5]).…”

Section: ⊓ ⊔mentioning

confidence: 99%

Classifying Groups With a Small Number of Subgroups

Betz

Nash²

2022

The American Mathematical Monthly

View full text Add to dashboard Cite

show abstract

“…Groups of order p 2 q were classified by Hölder (see [6], [3, p. 76], [4], or [1]). The previous lemma may also be justified by analyzing the cases described in these references.…”

Section: Solvable Groupsmentioning

confidence: 99%

Finite Groups Whose Intersection Graphs Are Planar

Kayacan¹,

Yaraneri²

2015

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H ∩K = 1 where 1 denotes the trivial subgroup of G. In this paper we characterize all finite groups whose intersection graphs are planar. Our methods are elementary. Among the graphs similar to the intersection graphs, we may count the subgroup lattice and the subgroup graph of a group, each of whose planarity was already considered before in [2,10,11,12].

show abstract

“…(iii) For the groups of order p q r, where p > q > r are primes, we use the list from Cole and Glover (1893). There they show that there is a normal subgroup s of order p and a normal subgroup H = s t of order pq.…”

Section: Groups Of the Type F M K Rmentioning

confidence: 99%

Fusions of Character Tables and Schur Rings of Abelian Groups

Humphries

Johnson

2008

Communications in Algebra

View full text Add to dashboard Cite

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. The theory is developed in terms of the S-rings of Schur and Wielandt. We discuss certain classes of pgroups which fuse from abelian groups and give examples of such groups which do not.We also show that a large class of simple groups do not fuse from abelian groups. The methods to show fusion include the use of extensions which are Camina pairs, but other techniques on S-rings are also developed.

show abstract

On Groups Whose Orders are Products of Three Prime Factors

Cited by 10 publications

References 0 publications

Classifying Groups With a Small Number of Subgroups

Classifying Groups With a Small Number of Subgroups

Finite Groups Whose Intersection Graphs Are Planar

Fusions of Character Tables and Schur Rings of Abelian Groups

Contact Info

Product

Resources

About