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Why are subgroups important when regarding groups? What information do they provide us? H of G

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closed as too broad by Matt Samuel, Henning Makholm, Adam Hughes, Shailesh, Moishe Kohan Dec 12 '16 at 5:03

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  • $\begingroup$ Sometimes a lot of information. For example, a finite group is cyclic iff it has one unique subgroup of every order dividing its order, and a group (in general) is solvable iff it has one normal subgroup such that it and the quotient group it determines are solvable...and many other things. Even simpler: in finite groups, the order of a subgroup stricly restricts the possible order of the whole group/ $\endgroup$ – DonAntonio Dec 11 '16 at 18:17
  • $\begingroup$ Also, every finite group is the subgroup of a permutation group. And don't forget the Abel-Ruffini theorem (there is no general formula for the roots of a quintic polynomial), which hinges on the theory of solvable groups. The definition of solvable groups makes heavy use of the concept of subgroups, and is, I'd venture to say, impossible to state without them. And the AR theorem is not a rare case. There are many theorems out there that hinge on the existence (or non-existence) of certain subgroups of given groups. $\endgroup$ – Arthur Dec 11 '16 at 18:23
  • $\begingroup$ Consider the group of isometries of the euclidean plane: they are things like translations, rotations, reflections, and combinations of these. Each of the things I just named -- the set of translations, the set of rotations about some fixed point, and the set of reflections (together with the identity map) -- constitute a subgroup. And each is important in its own way: translations preserve parallelism, for instance. $\endgroup$ – John Hughes Dec 11 '16 at 18:24
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I think the Sylow theorems give a good example of how studying subgroups of a given finite group can yield a better understanding of that group. For instance, some standard exercises in a group theory class include:

  • Show that no group of order $56$ is simple.

  • Show that every group of order $15$ is abelian.

And so forth. (Credit: I got those examples from Prof. Kimberly Elce's website)

There are more sophisticated examples of results along these lines, but this should already be enough to show that one use of subgroups is to aid in classifying groups.


I would also argue that studying subgroups is of independent interest. More generally, if I want to understand a class of mathematical structures (groups, rings, topological spaces, linear orderings, ...), then I also want to understand how objects in that class relate to each other: in particular, how can I build new ones from old ones? This question often splits into three sub-questions:

  • How can I transform one object into another?

  • How can I combine lots of objects into one big object?

  • How can I take one object, and "shrink" it to get another object?

In the context of groups, these correspond to (among other things) group homomorphisms, direct products, and subgroups. (And indeed, the HSP theorem in universal algebra shows that these operations are sufficeint to build a well-behaved class of structures from some "starting examples"!)

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