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Disproving transitivity

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This survey article is about proof techniques for or related to: subgroup metaproperty satisfaction
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A subgroup property p is termed a transitive subgroup property if whenever H \le K \le G are groups such that H satisfies property p in K and K satisfies property p in G, then H satisfies property p in G.

A subgroup property p is termed a t.i. subgroup property if it is transitive as well as identity-true: every group satisfies the property as a subgroup of itself.

This article discusses techniques used to prove that a given subgroup property is not transitive.

Also refer:

Contents

Quick discussion on transitivity and subordination

Given a subgroup property p, we define the subordination of p as the following property: H has the property in G if there exists an ascending chain of subgroups:

H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G,

such that each H_{i-1} satisfies property p in H_i. By definition, any group satisfies the subordination of p in itself, since we can take n = 0 and take a chain of length 0.

The subordination of any property is a t.i. subgroup property (it is both transitive and identity-true) and a t.i. subgroup property equals its own subordination.

Here are some quick points on the subordination operator:

  1. The subordination operator is an ascendant operator: If p is a subgroup property, p is stronger than its subordination.
  2. The subordination operator is a monotone operator: If p,q are subgroup properties such that p is stronger than q, then the subordination of p is stronger than the subordination of q.
  3. If p and q are subgroup properties such that q is t.i., then the subordination of p is stronger than q. In fact, the subordination of p is the strongest t.i. subgroup property among those weaker than q.

The basic proof idea

Suppose we want to prove that a given subgroup property q is not transitive. One approach is as follows: find an identity-true subgroup property psuch that p is stronger than q but the subordination of p is not stronger than q. Let's now see how such p can be chosen.

Using normality

The subordination of the property of being a normal subgroup is commonly called the property of being a subnormal subgroup. Thus, one way of showing that a given property q is not transitive is to show that every normal subgroup of a group satisfies q, but there are subnormal subgroups that do not satisfy q.

Here are some examples of subgroup properties that are satisfied by normal subgroups, but not satisfied by all subnormal subgroups, and hence are not transitive:

Using maximality

In a finite group, every subgroup is a submaximal subgroup -- there exists an ascending chain of subgroups starting from the subgroup and ending at the whole group such that each is a maximal subgroup of its successor. Thus, we have the following:

If an identity-true subgroup property q is satisfied by every maximal subgroup, but there is a subgroup of a finite group not satisfying it, then the subgroup property is not transitive. Examples include, for instance:

Looking at specific examples

This section discusses some general ideas for how to come up with specific examples to prove that a particular subgroup property is not transitive.

First, eliminate the kinds of groups where the property is transitive

To make the search for a counterexample efficient, it is helpful to first remove from consideration all groups where the property is transitive for obvious reasons.

General strategies:

  1. Suppose \alpha is a trim subgroup property: it is always satisfied by the whole group as a subgroup of itself, and by the trivial subgroup. Then, the trivial group cannot provide a counterexample. A group of prime order cannot provide a counterexample, and a group whose order is of the form p^2 or pq (where p,q are primes) cannot provide a counterexample. Thus, to start looking for groups where \alpha is not transitive, the place to start looking is groups of order p^3, p^2q, or pqr, where p,q,r are primes.
  2. Find properties \beta stronger than \alpha that are transitive, and eliminate all groups where the subgroups with property \alpha also have property \beta: For instance, the property of being a central subgroup (or more generally, a central factor) is transitive, and it is stronger than the property of being a normal subgroup. Thus, to look for examples of groups where normality is not transitive, we should look at groups where it is not true that all normal subgroups are central (or, where all normal subgroups are central factors). In particular, we should not look at abelian groups. We should also not look at simple groups, quasisimple groups, and characteristically simple groups.
  3. Try to find the precise left transiter for the property, and then show that the property does not imply its left transiter: The left transiter of a property \alpha is the unique property \beta such that H satisfies \beta in K if and only if, for every G containing K, whenever K satisfies \alpha in G, H satisfies \alpha in G. Showing that \alpha is not transitive is then equivalent to finding a subgroup H of a group K that satisfies \alpha but not \beta. Once we have such a pair, we find a G containing K such that K satisfies \alpha in G but H does not satisfy \alpha in G. An example of this is showing that normality is not transitive. One way of doing this is to show that left transiter of normal is characteristic, and combine this with the fact that normal not implies characteristic.

Selecting the better examples to begin with

As remarked above, we should start with groups of order p^3 when trying to prove that a trim subgroup property \alpha is not transitive. There are some general rules for what groups to prefer:

  • The cyclic group of order p^3 (and more generally, any cyclic group) is a very bad place to start looking. This is because subgroups of a cyclic group satisfy some very strong subgroup properties, and countering transitivity is in general extremely hard.
  • The other abelian groups of order p^3 (and more generally, of order p^n) are good places to start looking if there is no reason to believe that the property is transitive for all abelian groups. This includes the elementary abelian group and the product of the cyclic group of order p^2 and the cyclic group of order p.
  • Among the non-abelian p-groups, the quaternion group (exists only for p = 2) is a very bad place to start looking. This is because it has only one subgroup of order p, and this subgroup satisfies some very strong subgroup properties.
  • The dihedral group of order eight, and the two non-abelian groups for odd primes (prime-cube order group:U3p and prime-cube order group:p2byp) are good places to start looking.

Among groups of order p^2q and higher orders, here are some general ideas:

Some illustrations