Monatshefte ftir Mathematik 75, 333--337 (1971) C) by Springer-Verlag 1971
Intersection oI Chains ot Subgroups oI a Finite Group By
Said Sidki, Brasilia D.F., Brasil
(Received March 16, 1970) 1. Introduction Let G be a finite group and L ~-- L(G) be the lattice of subgroups of G. The interplay between the properties of G and those of L(G) has been studied in various ways and there are cases where characterization theorems have been obtained (see the standard reference, SUZUKI [5]). Therefore if a hybrid of some of the well-understood lattice properties is imposed on L(G), a description of the group G may be possible. The following question will disclose such a property. Consider the set R of maximal chains of L(G) connecting subgroups of G to E (the trivial subgroup). What could be said about G if the intersection of any two members of equal length of R were also a maximal chain ?
2. Definitions and Quoted Results We shall define the notions introduced in the question stated above along with other necessary concepts.
Definition 2.1. Let C be a subset of L. C is said to be a chain provided C is linearly ordered by inclusion; i. e., C -----{Ci; i ---- 0 . . . . . n} where C~ ~ C~+1 for i-~ 0 . . . . . n--1. We say C connects C0 to C~. If C~+1 is a maximal subgroup of C~ for all i, then C is said to be a maximal chain (or a path) of length n.
Definition 2.2. (See [4].) G is a CY-group provided L(G) satisfies the chained-Y condition: for any two maximal chains C = {C~}~, D----{D,~}~ of equal lengths connecting two distinct subgroups of G to E, C n D = {C~ a D~}~ is a maximal chain after all repetitions have been removed.
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Example 2.3. Consider G the dihedral group of order eight. The lattice of G is easy to determine and the CY-condition can be verified directly. Definition 2.4. (See [3].) G is an LY-group provided L(G) satisfies the long-Y condition: given A, B any two noncomparable subgroups of G, then L (A n B) is a chain (equivalently, A n B is a cyclic group of prime power order). Remark 2.5. For more general classes of groups see BAUMAN [1] and ISAACS [2].
Definition 2.6. G is a J-group ff L(G) satisfies the Jordan-Dedekind condition: given U and V, two subgroups of G, then any two maximal chains of subgroups that connect U to V have equal lengths. Theorem 2.7. [5, Theorem 9, p. 9.] A finite group G is a J-group i/
and only i] it is supersolvable. Definition 2.8. G is an LM-group provided L(G) is lower semimodular: for any two subgroups U and V of G, U n V is maximal in V whenever U is a maximal subgroup of U v V (the subgroup generated by U and V). Theorem 2.9. [5, Theorem 10, p. 10.] A finite group G is an LM-group i/and only if G is supersolvable and induces an automorphism o] prime order in each/actor group o/a principal series. 3. The Results
Let LY, J, CY, and LM denote the classes of LY-groups, J-groups, CY-groups, and LM-groups respectively. With the use of simple techniques the following inclusion diagram will be established:
LY
J
\\ // LYNJ
L
CY
/ LY N LM
/
LM
Intersection of Chains of Subgroups of a Finite Group
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A where I means B is contained in A. B
Lemma 3.1. Each o/the classes o/groups, CY, LY, Jo LM, is closed under transition to subgroups and homomorphic images. Theorem 8.2. CY is contained in J.
Pro@ Let G be a counterexample of minimal order. Let H be a proper subgroup of G of maximal order having two paths of unequal lengths connecting it to G. Let C:E=Co=Cl
c . . . cc
=H :c
+l : . . .
Cm=G
be a path of minimum length connecting E to H to G, and
D:E-----CocCI: ... c C ~ = H cD~+I c ... cD,,=-G be another path with n greater than m. Consider
D':Z=CocC~c...
cCk=HcD~+~c..,
c D m.
C n D' is a path that connects D mto E and its length is less than m; for Ck+l n D g + 1 = (~k - ~ H. Since D~ is a J-group, a contradiction is reached.
Remark 3.3. The previous theorem holds also for a group G where all paths between its subgroups are of finite length. The proof in this case is more involved. Theorem 3.4. CY is contained in LY.
Pro@ Let G be a CY-group. If U and V are two noncomparable subgroups of G and L( U n V) is not a chain, then U n V contains two noncomparable subgroups A and B. Since A n B is not a maximal subgroup of U n V, it is easy to see that the CY-eondition fails for G, Example 3.5. The following is a demonstration that CY 4= L Y n J. If T is a subset of a group G, denotes the subgroup of G generated by T. Let K be a cyclic group of prime order p and let this K be generated by an element a. Suppose K admits an automorphism b of order q3, q a prime; for instance in case p = 17, q =- 2. Let B be the group generated by b.
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Construct H, the semidirect product of K by B, and let these groups be naturally imbedded in H. Clearly H is supersolvable. Since the commutator subgroup of H is K, it follows that and c c ,
it is easy to see that H fails to be a CY-group; for n ~ and
n -----E.
Theorem 3.6. L M n L Y is contained in OY. Proo]. Let G be an L M , LY-group. Then, by Theorem 2.9, G is a J-group. Induct on the order of G. By Lemma 3.1 and induction, the maximal subgroups of G arc CY-groups. I t suffices to consider two maximal chains: C:Uo~ ... ~ C~=E
andD:D0~
. . . ~ D,---- B
where Co and D Oare maximal subgroups of G. Since G is an LM-group, N -----COn D Ois maximal in Co and Do. We may assume N to be different from C1 or D 1. Since L(N) is a chain we get C1 n N = D1 n N ---- C1 n D l, a maximal subgroup of Co n Do. This argument can be used to show Ct n D~ is maximal in Ci_l n D~_ 1 for all i. This completes the proof of the theorem. We conclude with a brief remark on a characterization of CY-groups. Remark 3.7. Using the methods and results of [1] and [2] it is possible to give a strong description of supersolvable LY-groups [3]. This result together with the theorem on finite LM-groups cited in section 2, can be used to characterize CY-groups as supersolvable LY-groups which do not allow H, the group of example 3.5, as a factor group
(see [E). References
[1] BAU~N, S.: Non-solvable /C-groups. Proc. Amer. Math. Soe. 15, 823--827 (1964). [2] I s l e s , I. M.: A note on /C-p-groups. Proe. Amer. Math. Soc. 17, 1451--1454 (1966).
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[3] SID~I, S. : A lattice property and its application to finite groups. A. M. S. Notices 14, 830 (1967). [4] SID~I, S.: Finite groups with chained-Y lattices of subgroups. A. M. S. .Notices 14, 941 (1967). [5] S~zuKi, M.: Structure of a Group ~nd the Structure of Its Lattices of Subgroups. Ergebnisse der Ma~hematik und ihrer Grenzgebiete, Heft 10. Berlin-G6ttingen-tteidelberg: Springer. 1956. Author's address 9 Dr. S~D SID~I Departamento de Mat~maticas Universidade de Brasflia Brasfli~ D. F., Brasil
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