Commutators and solvable groups

In summary, the conversation discusses commutators and how they relate to determining if a group is solvable or not. The Heisenberg group of 3x3 upper triangular matrices is an example of a two-step solvable group, but it is not clear how to show this. The commutator is used to show that the group GL(2,R) is not solvable. The conversation also discusses the concept of nilpotent and solvable groups, and gives an example of a solvable but not nilpotent group.
  • #1
JSG31883
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I am not really clear on what is meant by commutators. I know that the commutator of G is ABA^-1B^-1, but I am not sure how to check if a group is solvable by having the commutator eventually equal the trivial group.

For example, I know that the Heisenberg group of 3x3 upper triangular matrices is two-step solvable, but am not sure how to SHOW that. I know that it means that the first commutator doesn't equal the identity matrix and that the second one does... but how do I show this?

Also, how do I show that the group GL(2,R) (2x2 invertible matrices) IS NOT solvable?
 
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  • #2
you need to describe the commutator, so do it. take two arbitrary matrices in the upper triangular matrices and work out the commutator. see what happens. now take the commutator of this with another generic element of the comutator, see what you get. as it happens it is clearer to see that the lie algebra is solvable.


example: upper triangular 2x2 matrices with 1 on the diagonals, what is the commutator or two elements?

[tex]\left(\begin{array}{cc} 1&a\\0&1 \end{array}\right)\left(\begin{array}{cc} 1&b\\0&1 \end{array}\right)
\left(\begin{array}{cc} 1&-a\\0&1 \end{array}\right)
\left(\begin{array}{cc} 1&-b\\0&1 \end{array}\right)[/tex]

well?

as for the second example, GL contains SL which is simple.
 
  • #3
How can I show that the group G=<a,b,c> with [a,b]=b, [a,c]=1, [b,c]=1 is solvable but not nilpotent?

A group G is said to be nilpotent if G^i=identity for some i.
A group G is said to be solvable if it has subnormal series G=GncG3cG2cG1=identity... where all quotient groups are abelian.
 
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1. What is a commutator in group theory?

A commutator in group theory is an element of a group that measures the failure of two elements to commute. In other words, the commutator of two elements a and b is the element [a, b] = aba^-1b^-1. This measures how much a and b "cancel out" when multiplied together and then multiplied in reverse order.

2. How are commutators used to classify solvable groups?

Commutators are used to classify solvable groups by determining the number of times the commutator subgroup is contained in the group. Solvable groups are those that can be built up from abelian groups by taking successive quotients by commutator subgroups. The number of times the commutator subgroup is contained in the group is known as the "derived length" of the group, and it is used to classify solvable groups into different levels.

3. Can a non-solvable group have a commutator subgroup?

Yes, a non-solvable group can have a commutator subgroup. In fact, all groups have a commutator subgroup, which is defined as the subgroup generated by all commutators of the group. However, for non-solvable groups, the commutator subgroup will eventually reach the entire group, meaning it will have a derived length of 1. This is because non-solvable groups cannot be built up from abelian groups by taking successive quotients by commutator subgroups.

4. How do commutators relate to the concept of normal subgroups?

Commutators are closely related to the concept of normal subgroups. A normal subgroup is a subgroup that is invariant under conjugation by elements of the group. In other words, if H is a normal subgroup of G, then for any g in G, ghg^-1 is also in H. Commutators are a special case of normal subgroups, where the subgroup is generated by all commutators of the group. Therefore, commutators can be thought of as a type of normal subgroup.

5. Are all solvable groups also nilpotent?

Not all solvable groups are nilpotent. While all nilpotent groups are solvable, not all solvable groups are nilpotent. Nilpotent groups have a more restrictive definition, where the commutator subgroup eventually becomes the trivial group after successive quotients. Solvable groups, on the other hand, only require that the commutator subgroup eventually reaches the trivial group, but it may take multiple steps to get there. Therefore, there are solvable groups that are not nilpotent.

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