A survey article on some subgroup embeddings and local properties for soluble PST-groups
Let G be a group and p a prime number. G is said to be a Yp-group if whenever K is a p-subgroup of G then every subgroup of K is an S-permutable subgroup in NG(K). The group G is a soluble PST-group if and only if G is a Yp-group for all primes p. One of our purposes here is to define a number of...
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| Cite this: | A survey article on some subgroup embeddings and local properties for soluble PST-groups / J.C. Beidleman // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 197-203. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-1560212019-06-18T01:31:18Z A survey article on some subgroup embeddings and local properties for soluble PST-groups Beidleman, J.C. Let G be a group and p a prime number. G is said to be a Yp-group if whenever K is a p-subgroup of G then every subgroup of K is an S-permutable subgroup in NG(K). The group G is a soluble PST-group if and only if G is a Yp-group for all primes p. One of our purposes here is to define a number of local properties related to Yp which lead to several new characterizations of soluble PST-groups. Another purpose is to define several embedding subgroup properties which yield some new classes of soluble PST-groups. Such properties include weakly S-permutable subgroup, weakly semipermutable subgroup, and weakly seminormal subgroup. 2017 Article A survey article on some subgroup embeddings and local properties for soluble PST-groups / J.C. Beidleman // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 197-203. — Бібліогр.: 20 назв. — англ. 1726-3255 2010 MSC:20D10, 20D20, 20D35. http://dspace.nbuv.gov.ua/handle/123456789/156021 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let G be a group and p a prime number. G is said to be a Yp-group if whenever K is a p-subgroup of G then every subgroup of K is an S-permutable subgroup in NG(K). The group G is a soluble PST-group if and only if G is a Yp-group for all primes p.
One of our purposes here is to define a number of local properties related to Yp which lead to several new characterizations of soluble PST-groups. Another purpose is to define several embedding subgroup properties which yield some new classes of soluble PST-groups. Such properties include weakly S-permutable subgroup, weakly semipermutable subgroup, and weakly seminormal subgroup. |
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Beidleman, J.C. A survey article on some subgroup embeddings and local properties for soluble PST-groups Algebra and Discrete Mathematics |
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A survey article on some subgroup embeddings and local properties for soluble PST-groups |
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A survey article on some subgroup embeddings and local properties for soluble PST-groups |
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A survey article on some subgroup embeddings and local properties for soluble PST-groups |
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A survey article on some subgroup embeddings and local properties for soluble PST-groups |
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A survey article on some subgroup embeddings and local properties for soluble PST-groups |
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survey article on some subgroup embeddings and local properties for soluble pst-groups |
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Інститут прикладної математики і механіки НАН України |
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A survey article on some subgroup embeddings and local properties for soluble PST-groups / J.C. Beidleman // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 197-203. — Бібліогр.: 20 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT beidlemanjc asurveyarticleonsomesubgroupembeddingsandlocalpropertiesforsolublepstgroups AT beidlemanjc surveyarticleonsomesubgroupembeddingsandlocalpropertiesforsolublepstgroups |
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Algebra and Discrete Mathematics SURVEY ARTICLE
Volume 23 (2017). Number 2, pp. 197–203
c© Journal “Algebra and Discrete Mathematics”
A survey article on some subgroup embeddings
and local properties for soluble PST-groups
James C. Beidleman
Communicated by I. Ya. Subbotin
Abstract. Let G be a group and p a prime number. G
is said to be a Yp-group if whenever K is a p-subgroup of G then
every subgroup of K is an S-permutable subgroup in NG(K). The
group G is a soluble PST-group if and only if G is a Yp-group for
all primes p.
One of our purposes here is to define a number of local proper-
ties related to Yp which lead to several new characterizations of
soluble PST-groups. Another purpose is to define several embed-
ding subgroup properties which yield some new classes of soluble
PST-groups. Such properties include weakly S-permutable subgroup,
weakly semipermutable subgroup, and weakly seminormal subgroup.
1. Introduction and statement of results
All groups considered in this survey are finite.
A subgroup H of a group G is said to permute with a subgroup K
of G if HK is a subgroup of G. H is said to be permutable (S-permutable)
if it permutes with all the subgroups (Sylow subgroups, respectively) of G.
Examples of permutable subgroups include the normal subgroups of G.
However, if G is a modular, non-Dedekind p-group, p a prime, we see
permutability is quite different from normality. For instance, letting Cn
denote the cyclic group of order n, we see that C2 is permutable but not
2010 MSC: 20D10, 20D20, 20D35.
Key words and phrases: S-permutable subgroup, semipermutable subgroup,
seminormal subgroup, PST-group.
198 On some subgroup embeddings
normal in the group C8 ⋊C2 where the generator for C2 maps a generator
of C8 to its fifth power. Kegel [12] proved that an S-permutable subgroup
is always subnormal. In particular, a permutable subgroup of a group is
subnormal.
A group G is called a PST-group (PT-group) if S-permutability (per-
mutability, respectively) is a transitive relation. By Kegel’s result, a
group G is a PST-group (PT-group) if every subnormal subgroup of G is
S-permutable (permutable, respectively) in G.
A number of research papers have been written on these groups. See,
for example, [1–10,16].
Another class of groups is the so called T-groups. A group G is a
T-group if normality in G is transitive, that is, if H E K E G then H E G.
There are several nice characterizations of T-groups in [15].
Soluble PST-, PT- and T-groups were characterized by Agrawal [1],
Zacher [18] and Gaschütz [11] respectively.
Theorem 1. 1) A soluble group G is a PST-group if and only if the
nilpotent residual L of G is an abelian Hall subgroup of G on which
G acts by conjugation as a group of power automorphisms.
2) A soluble PST-group G is a PT-group (T-group) if and only if G/L
is a modular (Dedekind, respectively) group.
Note that if G is a soluble T-, PT- or PST-group then every subgroup
and every quotient of G inherits the same properties.
We mention that in [6, Chapter 2] many of the beautiful results on
these classes of groups are presented.
Subgroup embedding properties closely related to permutability and
S-permutability are semipermutability and S-semipermutability. A sub-
group X of a group G is said to be semipermutable (S-semipermutable)
in G provided that it permutes with every subgroup (Sylow subgroup,
respectively) K of G such that gcd (|X|, |K|) = 1. A semipermutable
subgroup of a group need not be subnormal. For example, a 2-Sylow
subgroup of the non-abelian group of order 6 is semipermutable but not
subnormal.
Note that a subnormal semipermutable (S-semipermutable) subgroup
X of a group G must be normalized by every subgroup (Sylow subgroup,
respectively) P of G such that gcd (|X|, |P |) = 1. This observation was the
basis for Beidleman and Ragland [10] to introduce the following subgroup
embedding properties.
J. C. Beidleman 199
A subgroup X of a group G is said to be seminormal (S-seminormal)1
in G if it is normalized by every subgroup (Sylow subgroup, respectively)
K of G such that gcd (|X|, |K|) = 1.
By [10, Theorem 1.2], a subgroup of a group is seminormal if and
only if it is S-seminormal. Furthermore, seminormal subgroups are not
necessarily subnormal; it is enough to consider a non-subnormal subgroup
H of a group G such that π(H) = π(G). To see some of the properties
of these subgroups see Examples 1, 2 and 3 in Section 2. However, a p-
subgroup of a group G, p a prime, which is also seminormal is subnormal
([10, Theorem 1.3]).
Semipermutable, S-semipermutable and seminormal subgroups have
been investigated in [10,17,19,20].
The following result gives some embedding properties on subnormal
subgroups of a group which yields several characterizations of soluble
PST-groups.
Theorem 2 ([10, Theorem 1.5]). Let G be a soluble group. Then the
following statements are pairwise equivalent:
1) G is a PST-group;
2) all the subnormal subgroups of G are seminormal in G;
3) all the subnormal subgroups of G are semipermutable in G;
4) all the subnormal subgroups of G are S-semipermutable in G.
The following is a beautiful result of H. Wielandt which seems to have
inspired the authors of [5] to introduce the concept of weakly S-permutable
subgroups of a subgroup H of a group G.
Theorem 3 ([13, Theorem 7.3.3]). Let H be a subgroup of a group G.
Then the following statements are equivalent:
1) H is subnormal in G;
2) H is subnormal in 〈H, Hg〉 for all g ∈ G;
3) H is subnormal in 〈H, g〉 for all g ∈ G.
This embedding property led to several new characterizations of soluble
PST-groups which are presented in the following theorem from [5].
Theorem 4 ([5]). Let G be a group. The following statements are pairwise
equivalent:
1) G is a soluble PST-group.
2) Every subgroup of G is weakly S-permutable in G.
1Note that the term seminormal has several different meanings in the literature.
200 On some subgroup embeddings
3) For every prime number p, every p-subgroup of G is weakly S-
permutable in G.
Theorems 3 and 4 motivate the following definition.
Definition 1. Let H be a subgroup of a group G.
1) H is said to be weakly S-permutable in G if whenever g ∈ G and
H is S-permutable in 〈H, Hg〉, then H is S-permutable in 〈H, g〉.
2) H is said to be weakly semipermutable in G if whenever g ∈ G
and H is semipermutable in 〈H, Hg〉, then H is semipermutable
in 〈H, g〉.
3) H is said to be weakly S-semipermutable in G if whenever g ∈ G
and H is S-semipermutable in 〈H, Hg〉, then H is S-semipermutable
in 〈H, g〉.
4) H is said to be weakly seminormal in G if whenever g ∈ G and H
is seminormal in 〈H, Hg〉, then H is seminormal in 〈H, g〉.
The next theorem relates the concept of S-permutable subgroups of a
group G with weakly S-permutable subgroups of G.
Theorem 5 ([5]). A subgroup H of a group G is S-permutable in G if
and only if H is S-permutable in 〈H, g〉 for every g ∈ G.
Theorem 5 and its proof are used to establish Theorem 6 in [7].
Theorem 6 ([7]). Let H be a subnormal subgroup of a group G. Then
1) H is S-semipermutable in G if and only if H is S-semipermutable
in 〈H, g〉 for every g ∈ G.
2) H is seminormal in G if and only if H is seminormal in 〈H, g〉 for
every g ∈ G.
A class of groups G is a PST-group if and only if Sylow permutability
is a transitive relation in G.
We next define several local properties which provide a number of
new local characterizations of soluble PST-groups.
Definition 2. Let G be a group and p be a prime. Then
1) G is a Yp-group if for every p-subgroup K of G every subgroup of K
is S-permutable in NG(K).
2) G is a Ŷp-group if for every p-subgroup K of G every subgroup of K
is semipermutable in NG(K).
3) G is a Ỹp-group if for every p-subgroup K of G every subgroup of K
is S-semipermutable in NG(K).
J. C. Beidleman 201
4) G is a ˜̃Yp-group if for every p-subgroup K of G every subgroup of K
is seminormal in NG(K).
5) G is a Y p-group if for every p-subgroup K of G every subgroup
of K is weakly S-permutable in NG(K).
6) G is a Ỹ p-group if for every p-subgroup K of G every subgroup
of K is weakly S-semipermutable in NG(K).
7) G is a ˜̃Y p-group if for every p-subgroup K of G every subgroup
of K is weakly seminormal in NG(K).
The following result is a very nice local characterization of soluble
PST-groups.
Theorem 7 ([6, Theorem 2.2.9] and [4, Theorem 4]). A group G is a
soluble PST-group if and only if it satisfies Yp for all primes p.
Our next result shows how some of the classes in Definition 2 are
related to the class Yp.
Theorem 8 ([10, Theorem 1.8]). Let p be a prime and G a group. Then
Yp = Ŷp = Ỹp = ˜̃Yp.
Using Theorems 7 and 8 we note that the next result shows all of the
classes Y p, Ỹ p and ˜̃Y p are just Yp.
Theorem 9 ([7]). Let p be a prime and G a group. Then
1) G ∈ Yp if and only if G ∈ Y p.
2) G ∈ Ỹp if and only if G ∈ Ỹ p.
3) G ∈ ˜̃Yp if and only if G ∈ ˜̃Y p.
From Theorems 8 and 9 we obtain several results that yield new local
characterizations of soluble PST-groups.
Corollary 1. Let p be a prime. Then
Yp = Y p = Ŷp = Ỹp = Ỹ p = ˜̃Yp = ˜̃Y p.
Using Theorem 9 and Corollary 1 we obtain one of the main results
of this survey paper.
Theorem 10 ([7]). Let G be a group and p a prime. Then the following
statements are pairwise equivalent:
1) G is a soluble PST-group.
2) G is a Yp-group for all primes p.
202 On some subgroup embeddings
3) G is a Y p-group for all primes p.
4) G is a Ŷp-group for all primes p.
5) G is a Ỹp-group for all primes p.
6) G is a Ỹ p-group for all primes p.
7) G is a ˜̃Yp-group for all primes p.
8) G is a ˜̃Y p-group for all primes p.
2. Examples
Example 1. Let S4, A4 and K4 denote, respectively, the symmetric group
of order 4, the alternating group of order 4, and the Klein 4-group. Let
G = S4 and let H = 〈(123)〉. The H is S-semipermutable in G but it is
not semipermutable in G since it does not permute with an element of
order 2 in K4, the Sylow 2-subgroup of A4.
An S-permutable subgroup of a group is subnormal. That this is not
the case with S-semipermutable subgroups can be seen in the subgroup H
in S4. Notice that H is not seminormal in S4.
Example 2. Let D10 = 〈x, y | x5 = y2 = 1, xy = x−1〉, the dihedral
group of order 10, and C15 = 〈t, s | t5 = s3 = 1, ts = st〉, the cyclic group
of order 15. Let G = D10 ×C15 and let K = 〈y〉×〈t〉. Since 〈s〉 centralizes
K it follows that K is seminormal in G. Note that K is not subnormal
in G.
Example 3. Let H = 〈x〉⋊ 〈y〉 be a semidirect product of a cyclic group,
〈x〉, of order 11 by a cyclic group, 〈y〉, of order 5. Let G = H × S4. Set
K = 〈x〉×S3 where S3 is a copy of the symmetric group on three elements
in S4. Then K is a seminormal subgroup of G which is not subnormal.
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Contact information
J. C. Beidleman Department of Mathematics,
University of Kentucky,
715 Patterson Office Tower,
Lexington, KY (USA)
E-Mail(s): james.beidleman@uky.edu
Received by the editors: 07.01.2017.
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