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Let $G$ be a finite group. I will say that a set of a subgroups $H_1,\ldots ,H_k$ defines a basis for a group $G$ if any subgroup $H$ of $G$ there exists $S\subset [k]$ such that $H=\cap_{i\in S}H_i$.

My question is does it possible to give any upper bounds on the size of the minimal basis for $G$. For example does it possible to prove that for any $G$ there exists a basis of size at most $|G|^{10}$?

For example for an Abalian group it is easy to show that there is always exists a basis of size $|G|$. For example in case $A=Z_p^n$ it will be all maximal subgroups.

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On the abelian case: The maximal subgroups do not form a basis. Think of a cyclic group of prime power order, where the subgroups form a chain and hence you need all of them in a basis (though I think your upper bound would still hold). –  Florian Eisele Aug 14 '12 at 9:56
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Evidently your basis must contain all maximal subgroups. A little googling finds that Liebeck, Praeger and Saxl have a result describing almost all the maximal subgroups of the symmetric and alternating groups. The description is rather complicated, but you could probably use it to get a lower bound. –  HJRW Aug 14 '12 at 9:57
    
To Florian Eisele: thanks for a comment I made a change. –  Klim Efremenko Aug 14 '12 at 10:11
    
The name basis of a group I invented just now, may be this notion have an other name in the literature. –  Klim Efremenko Aug 14 '12 at 10:15
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Why the tags "lie-groups" and "rt.representation-theory"? –  Frieder Ladisch Aug 14 '12 at 12:40
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up vote 6 down vote accepted

This is an extended comment. An element of a lattice is called meet-irreducible if it cannot be expressed as a meet of two elements containing it. For a finite lattice, the meet irreducible elements are the unique minimal generating set for the lattice under meet. Your bases are exactly meet generating subsets of the subgroup lattice and the unique minimal basis is the set of meet irreducible subgroups (sometimes called primitive subgroups in the literature). It is known that the minimal degree of a faithful permutation representation of G is a lower bound for the size of this set. I am not sure what upper bounds are known.

You might want to look at Johnson, D. L. Minimal permutation representations of finite groups. Amer. J. Math. 93 (1971), 857–866.

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Yes. Fixed...... –  Benjamin Steinberg Aug 14 '12 at 21:07
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Thanks for your comment. In fact I even do not know how to prove that the number of maximal subgroups is at most $|G|$ –  Klim Efremenko Aug 14 '12 at 22:09
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You should talk to a specialist in subgroup latticed. Maybe Shareshian. –  Benjamin Steinberg Aug 14 '12 at 22:31
    
Thanks. I will accept your answer. And ask the question again in the form what is the maximal number of maximal subgroups. –  Klim Efremenko Aug 15 '12 at 12:26
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