I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from there I learned about composition series and the Jordan-Holder theorem. I was still unclear on how this relates to my problem, so I did some digging into short exact sequences. I read this paper, http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/splittinggp.pdf , and I understand everything in it, but I am still unclear on how this relates to the extension problem. I would love to understand how we use Jordan-Holder theorem in conjunction with short exact sequences. It would be great if anyone had some links to any other papers that might pick up where aforementioned paper left off. I have tried to find such papers on the internet but they all seem to be way more advanced then that Conrad paper above or they have no further information than said paper. I know there are similar questions on math.stackexchange but none of them have the information I am looking for.
2026-04-06 19:03:07.1775502187
how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?
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