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spamiam
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What are some simple groups that have non-normal subgroups? The only example I can think of is the alternating group for n > 4.
mathwonk said:all non abelian simple groups have non normal subgroups. i.e. they have even order...
...so they have elements of order 2, hence subgroups of order 2, which are necessarily non normal.
spamiam said:Wait, all non-abelian simple groups have even order? Or did you mean that as an example?
Okay, the element of order 2 follows from Cauchy's theorem, but why are these subgroups "necessarily" non-normal? Is it just because we were already assuming the group was simple or is there a deeper reason?
I was kind of hoping for some specific examples of known groups. The only examples of simple groups with which I'm familiar are [itex] \mathbb{Z}/p\mathbb{Z}[/itex] and [itex]A_n[/itex] for n > 4. Are there any other well-known ones?
Thanks again!
micromass said:This is the contents of the celebrated Feit-Thompson theorem.
To my knowledge, example of simple groups are kind of tricky. The article http://en.wikipedia.org/wiki/List_of_finite_simple_groups gives a list of simple groups.
A non-normal subgroup in a simple group is a subgroup that is not a normal subgroup, meaning it does not satisfy the condition that every element of the subgroup commutes with every element of the group. In other words, there exists at least one element in the subgroup that does not commute with an element in the group.
Non-normal subgroups play a crucial role in the structure of simple groups. They help identify certain properties and characteristics of the group, and can be used to classify different types of simple groups.
Yes, a simple group can have multiple non-normal subgroups. However, in order for a simple group to have more than one non-normal subgroup, it must have a large number of elements and a complex structure.
Non-normal subgroups are the complement of normal subgroups in simple groups. This means that in a simple group, every subgroup that is not normal is considered a non-normal subgroup.
Examples of non-normal subgroups in simple groups include the alternating group An, where n ≥ 5, and the Mathieu group M11. These are examples of non-normal subgroups that have been extensively studied in the field of group theory.