Welcome to NRICH.

 
Finding subgroups of a group
By Chris Park on Sunday, June 02, 2002 - 08:34 am:

Hey guys, I'm just having a few queries while doing this stuff.

I've come across an exercise which asks you to find the proper subgroups of a given set which forms a group. Do you have to write up an operation table to do this? If not, how else can you approach such a question.

Plus, theres other exercises involving isomorphisms etc. Do we have to write up operation tables up for these to?

Writing up the operation tables is so tedious!

By Ian Short on Monday, June 03, 2002 - 10:57 am:

Hi, writing up operation tables is only really useful for groups of small order.

One of the most useful tools is Lagrange's theorem, which states that for a finite group G, the order of a subgroup H must divide the order of G.

Lagrange's theorem can be utilised in various ways.

For instance, if G has prime order p then it has no proper subgroups since only 1 and p divide p.

Further, note that if x is an element of G of order n then [1,x,x2,..,xn-1]is a subgroup of G order n. Hence applying Lagrange to H, we see that if x is in H then n | |H|. This helps to find all possible subgroups of a given order.

Tell me the particular group you're dealing with. I don't recall ever writing up operation tables for groups of order larger than 6.

Ian

By Chris Park on Monday, June 03, 2002 - 01:12 pm:

Could you explain what you were saying in your second last paragraph, it just doesnt make sense yet.

Instead of me giving you an example, could you explain to me the methods of finding subgroups of a given group. Especially of those groups with higher orders than 6 or so.

Thanks

By Ian Short on Tuesday, June 04, 2002 - 10:43 am:

Yes, the 2nd to last paragraph is no good at all.

The only point I was trying to make was that the order of an element divides the order of a group (since if xn=1 and n is smallest such, then [1,x,x2,..,xn-1] is a subgroup of G with order n therefore n | |G|). Does that make sense? How much group theory have you done before?

Finding all subgroups of a given group G is a decidedly non-trivial exercise. There are no simple general methods and usually subgroups are identified utilising a variety of different results.

Lagrange's theorem is the most fundamental result.
A more advanced, but very useful technique comes through using Sylow's theorems. For instance, one of these theorems says that if p a prime and pr | |G| then there is a subgroup of G of order ps (s £r). I could state Sylow's theorems in full generality, but the proofs are long.

So finding subgroups becomes easier when you know more group theory and hence can apply different techniques. Here are some pointers though (with Lagrange's theorem):

(1) Prime order groups have no proper subgroups, as previously noted.

(2) Groups order pq (p,q prime) have proper subgroups of orders p and q, but their may be lots of them. (Similarly groups orders with not many prime factors are more tractable).

(3) There is another theorem (proven from Lagrange) that the only groups of order 2p (for p prime) are the 'cyclic' and 'dihedral' groups (there're only 2 possibilities). These groups are respectively all possible rotation and rotations+reflections of a p-gon (rotations and reflections that fix the p-gon). Are these groups familiar?

(4) Bear in mind that if A and B are groups then AxB (A cross B) is also a group, so this is an efficient way of describing groups.

(5) Note that all even order groups must contain an element of order 2. Otherwise we could pair off elements of the group with their inverse (x and x-1 distinct when x not identity as not order 2) giving an odd number of elements in total (lots of pairs plus single identity).


Here I have indicated some brief ideas. Sylow theorems are not examined in Cambridge until the 3rd year for undergraduates (proofs long), just to indicate the level we're looking at.

Tell me if you want more information. An eminent group theorist in Cambridge said to me that he used to, "love calculating all groups of small orders when he was an undergraduate" (slightly different question, but I keep thinking of it). Generally integers with lots of prime factors are harder.

Ian