Finding groups by semidirect products

• Jösus
In summary, the speaker is studying group theory on their own and has reached a roadblock at order 16. They are seeking help and have been exploring Sylow Theory, normal subgroups, and semidirect products. They are looking for some guidance and resources to better understand the concept and solve their problem. They have been provided with a link to a classification of groups.
Jösus
Hello

Lately, I have been studying some group theory. On my own, I should add, so I don't really have any professor (or other knowledgeable person for that matter) to ask when a problem arises; which is why I am here.

I had set out to find all small groups (up to order 30 or something), up to isomorphism. Up until the unique group of order 15, no problem arises. However, at order 16 I have gotten stuck. Naturally, the abelian ones poses no real challange, but as only one prime is involved, Sylow Theory seems to be of little help, and so seems any other method I have tried. I would greatly appreciate some small hints on how to proceed, but I would like to do the rest myself. I am in the process of dropping my goal of finding the groups of said orders, but this one has been bugging me. So has the next one, but there I at least think I got off to a decent start.
The next problematic order is 21 (and a whole bunch of others with similar factorisation into primes). I try to approach it as follows: Sylow Theory tells us that if G is a finite group, p divides the order of G and p is a prime, then $$|Syl_{p}(G)| \equiv 1 (mod \: p)$$ and $$|Syl_{p}(G)|$$ divides $$|G|$$. From this, and the fact that 3 is congruent to 3 modulo 7, we draw the conclusion that any group of order 21 contains a normal subgroup of order 7. This subgroup is, as it is of prime order, a cyclic one. Moreover, the intersection of this normal subgroup and any Sylow 3-subgroup is simply the singleton $$\{1_{G}\}$$. Also, if we by S and T denote the groups of order 3 and 7, respectively, we have $$ST = \{st | s \in S, t \in T\} = G$$. Thus G is the semidirect product $$T \rtimes S$$. Now, as the involved groups are cyclic, this seems to be an easy match from here on. Here I guess I should give a presentation for the group, and it would be something like $$G = \langle a, b | a^7 = b^3 = 1, b^{-1}ab = a^k \rangle$$ where k is one of the integers 1, ... ,6. k = 1 yields the abelian group (the cyclic) of order 21. Another one would be the group with presentation $$\langle a, b | a^7 = b^3 = 1, b^{-1}ab = a^2 \rangle$$.

Here I am stuck. There seems to be too many possibilities for it to be reasonable to just play around with the relations and try to look for equalities between the different products. I know, I should really read more about presentations, but I don't have any good litterature for the moment. If anyone would feel like helping out, tell me if I made some terrible mistakes and faulty assumptions or anything else that could help me get a grasp on what I am doing, I would be very thankful.

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Doing a little reading, it seems that any semidirect product of N and H with N normal in G arises from a homomorphism from H into the automorphism group of N. I guess it's too much to hope for that each homomorphism determines a distinct isomorphism class, but at least it cuts things down somewhat.

First of all; thank you for taking the time to read and post. However, when thinking about this problem a bit more, it seems like my problems goes deeper than I first though. I simply haven't understood the concept well enough yet, or at least, I still find it a bit too technical to be able to produce results from what I know. I shall sit down and read some more, and then give it another shot!

What is a semidirect product in group theory?

A semidirect product is a mathematical construction used in group theory to combine two groups together. It is denoted by the symbol ⋈ and is defined by a certain type of group action between the two groups. This construction allows us to create new groups with unique properties from existing groups.

How do you find groups by semidirect products?

To find groups by semidirect products, you need to have two groups with compatible actions. This means that one group must act on the other group in a specific way. Once you have identified these groups, you can use the semidirect product construction to combine them and create a new group. This process involves taking the direct product of the two groups and then modifying it by the group action.

What are some applications of finding groups by semidirect products?

Finding groups by semidirect products has various applications in mathematics and physics. It can be used to study symmetry in geometric objects, such as crystals, and to classify finite groups. In physics, it has applications in gauge theory and the study of particle interactions.

What are some properties of groups obtained by semidirect products?

Groups obtained by semidirect products have some interesting properties. For example, if the two groups used in the construction are both Abelian, then the resulting group will also be Abelian. Additionally, the order of the resulting group is equal to the product of the orders of the two original groups.

Are there any limitations to finding groups by semidirect products?

While finding groups by semidirect products is a powerful tool, it cannot be used for every pair of groups. The group actions must be compatible, and not all groups have compatible actions. Additionally, the resulting groups may not always have desirable properties, such as being simple or solvable.

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