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I'm taking an algebraic structures class and we are doing a lot of work involving group theory. Specifically, dihedral groups, abelian groups, isomorphisms, cyclic groups, and others. I'm finding it interesting but I'm lost on what the point of it all is. Is it just something that mathematicians like to play around with or does it have any real-world applications? Can we use it practically?

I understand the general applications of permutation groups and the dihedral group, but what about something more specific? Such as, for what reason is it necessary to know that a group is cyclic?

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    $\begingroup$ Did you consider checking out Wikipedia's page on "Group Theory" and looking in the heading "Applications of Group Theory", which includes Physics and the role they play in the standard model, Chemistry, and Cryptography? Or a quick google search, which would lead you to this Keith Conrad post ? $\endgroup$
    – Arturo Magidin
    Commented Apr 5, 2021 at 21:23
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    $\begingroup$ See this question with 8 answers already in the case of finite groups. $\endgroup$
    – Moishe Kohan
    Commented Apr 5, 2021 at 21:29

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Of course we can use it practically! Group theory is actually one of the most useful algebraic structures outside of mathematics. It has a lot of applications in physics (quantum mechanics), chemistry (study the symmetry of molecules), cryptography (some cryptosystems, like the El-Gamal, are based on the structure of a certain group), and also pops up in other branches of mathematics that may seem unrelated to algebra at first glance. All those applications stem from the fact that group theory, at its core, studies symmetries of objects, and symmetries play a big role in many branches of science.

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