Proving G is Solvable or Finding a Counter-Example

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In summary, the conversation is discussing the solvability of a group G with a solvable maximal normal subgroup. The group G/H is simple but not necessarily abelian, leading to the possibility of a counter-example. The examples of symmetric group S_5 and quaternion group are mentioned, but they do not work as a counter-example. The possibility of G being a direct product of a solvable group is brought up as a potential counter-example.
  • #1
burritoloco
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Homework Statement


Suppose G has a solvable maximal normal subgroup. Either prove G is solvable, or give a counter-example.


Homework Equations





The Attempt at a Solution


We have that G/H is simple, not necessarily abelian, so my guess is there could be a counter-example. However, I'm not sure what it is.
I know the symmetric group S_5 is not solvable and it has the unique proper normal subgroup A_5, but A_5 is not solvable, so this doesn't work as a counter.
Moreover, the quaternion group, dihedral groups are solvable too. Any help please?
My exam is tomorrow... Thanks for the help!
 
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  • #2
What if G is the direct product of a solvable group with something suitable? Can this give you a counterexample?
 

What does it mean for a group G to be solvable?

In group theory, a group G is considered solvable if there exists a sequence of subgroups starting with the trivial subgroup and ending with the whole group, such that each subgroup is a normal subgroup of the previous one and the factor groups are all abelian.

How do you prove that a group G is solvable?

One way to prove that a group G is solvable is by using the concept of normal series. If we can construct a normal series for G with abelian factor groups, then G is solvable. Another method is by showing that G has a normal subgroup with an abelian quotient group, and then repeating this process until we reach the trivial subgroup.

What is a counter-example in the context of proving the solvability of a group?

A counter-example is an example that disproves a statement or theory. In the context of proving the solvability of a group G, a counter-example would be a group that does not meet the criteria for being solvable, thus disproving the statement that all groups are solvable.

Can a group be both solvable and unsolvable?

No, a group cannot be both solvable and unsolvable. A group is either solvable or unsolvable, and this is determined by its structure and the properties of its subgroups. A group cannot have contradictory properties.

What are some real-world applications of finding counter-examples in group theory?

Finding counter-examples in group theory can help us understand the limitations and exceptions to certain theories and concepts. This is important in fields such as cryptography, where the security of certain algorithms relies on the properties of groups. By finding counter-examples, we can identify weaknesses in these algorithms and improve their security. Counter-examples can also help us discover new properties and patterns in groups, leading to advancements in various fields of mathematics and science.

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