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I would like to know if every finite group is an extension of some group by another. Thanks

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Every group is an extension of itself by the trivial group. Or, do you mean something else? –  Zev Chonoles Jun 8 '12 at 20:50
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Or of the trivial group by itself. =] –  Tara B Jun 8 '12 at 20:51
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The group with one element is an extension of itself by itself, but that is not very interesting. A group with two elements is an extension of itself by a group of one element, but that is still not very interesting. To be an interesting extension, it must have a normal subgroup that has more than one element, but not all of the elements. The groups that are not interesting extensions are called simple groups. –  Jack Schmidt Jun 8 '12 at 20:51
    
@Jack: I think that would probably do for an answer (perhaps including an example of a simple group). –  Tara B Jun 8 '12 at 20:51

1 Answer 1

The group with one element is an extension of itself by itself, but that is not very interesting. A group with two elements is an extension of itself by a group of one element, but that is still not very interesting. To be an interesting extension, it must have a normal subgroup that has more than one element, but not all of the elements. The groups that are not interesting extensions are called simple groups, though usually the group with one element is not considered simple, but "trivial".

For example the additive groups $\mathbb{Z}/p\mathbb{Z}$ of integers mod p, where p is a prime are all simple groups.

The orientation-preserving symmetries of a regular icosahedron are a simple group of order 60.

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