# Is every finite group an extension?

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
Or of the trivial group by itself. =] –  Tara B Jun 8 '12 at 20:51
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

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