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I want to understand $D_4 \times \mathbb{Z}_2$ and I haven't found too much information about it online or in my books. There's a few things I'd like to know.

  • Is there any particular way of going about understanding it in comparison to other groups? For example, how does it differ from any other group of order $16$ (like $V_4^2$, $D_8$, $\mathbb{Z}_2^4$ or $V_4 \times D_2$)?

  • Has any mathematician ever produced a list of properties of a given group that one should look at first in order to have a better intuitive understanding of the structure of said group?

  • How would you go about "getting" this group? I understand this is largely opinion based but I believe it could give me a better grasp of how to deal with abstract groups.

I'd also be interested in book recommendations (especially from authors that focus on particular examples of groups found in the real world and in other subjects) or topics in mathematics that I could look into to understand groups better.

William A. Noble
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    Given that it is a direct product, the easiest way to start is to understand the factors: $D_4$ (by which you seem to mean the dihedral group of order $8$) and $\mathbb{Z}_2$. The latter is very easy to understand, and you may already be familiar with the former. – Arturo Magidin Aug 06 '21 at 06:06
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    Often it is easy to distinguish groups of small order by some invariants, e.g., by the order, by its center, or commutator, or by the possible orders of elements, or by solvability or nilpotency. Fo example, $\Bbb Z_2^4$ is abelian, whereas $D_4\times \Bbb Z_2$ is not. – Dietrich Burde Aug 06 '21 at 09:43

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For groups of small order $n$ (as in, $n\le 30$ or so), it is tedious but relatively straightforward to figure out all possible groups of order $n$. You need to know Sylow and Galois theory as well as semidirect products. I'm not sure if these topics are familiar to you, but you might learn them in a typical second or third semester algebra class.

It actually turns out that groups of order $2^n$ are abundant. Here is a summary groups of order $\le 20$, for example:

  • there is one group of order $1,2,3,5,7,11,13,15,17$ and $19$
    • (note that $15$ is the only number here which isn't prime or $1$)
  • there are two groups of order $4,6,9,10$, and $14$
    • for orders $4,6,10$, and $14$, these are just the cyclic and dihedral groups
  • there are five groups of order $8,12,18$, and $20$ (here it gets interesting)
  • there are fourteen groups of order $16$
  • the next $n$ for which there are more than $5$ groups occurs at $n=24$. There are $15$ groups of order $24$.

The rough picture which emerges is "more prime divisors means more groups." If you have one prime divisor, i.e., a group $G$ of prime order $p$, then $G$ is necessarily cyclic. So there can only be one group with order $2,3,5,7$, etc.

If $G$ has order $pq$ where $p$ and $q$ are both prime, there are still few possibilities. Most of the time, Sylow theory forces $G$ to be a product $\Bbb Z_p\times\Bbb Z_q$, which is just the cyclic group of order $pq$. In any other case, $G$ is a "twisted product" of $\Bbb Z_p$ and $\Bbb Z_q$, and the isomorphism class depends on the action of $\Bbb Z_p$ on $\Bbb Z_q$ (where $p>q$).

If $G$ has order $pqr$ with $p,q,r$ prime, it gets even more complicated, though quite a bit can be said (eg here).

Anyway, this might all be overkill for your question. My point is just that the structure required of groups imposes enough restrictions that we CAN figure out all groups of a given small order. (For larger order groups, this gets hard FAST.)

Here are more observations that help:

  1. Whenever $G$ is a product of two other groups $H$, $K$, understanding $G$ is exactly the same as understanding both $H$ and $K$.
  2. The structure of abelian groups is relatively easy to understand because they are products of cyclic groups.

So if you have a group of order $16$ like $G=D_4\times\Bbb Z_2$, then if you understand $D_4$ and $\Bbb Z_2$ there is really nothing more to say about $G$. The product structure means that $D_4$ and $\Bbb Z_2$ don't really "interact" ever: we just have $(x,y)(x',y')=(xx',yy')$ for anny $x,x'\in D_4$, $y,y'\in\Bbb Z_2$, and $(x,y),(x',y')\in G$.

Now you only need to understand $D_4$ (which is one of the only two non-abelian groups of order $8).

I can't think of any references about groups appearing in nature, but understanding symmetries of objects is relevant in a ton of applied science including chemistry (enantiomers/chirality), physics (rotation groups and Lie groups), and computer science (Hamming weights). As an undergrad I found Gallian's book "Contemporary Abstract Algebra" accessible and interesting, especially some of the special topics towards the end.

pancini
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  • Thank you for the book recommendation and the long answer. I didn't mean groups found in nature but rather any concrete example of where a group might pop up. The Rubik's Cube group, for example. As to $D_4 \times \mathbb{Z}_2$, what I really wanted was an intuition on the inner workings of it, but I think now that my question is a bit poorly written and vague. I might delete it. – William A. Noble Aug 06 '21 at 08:54
  • @WilliamA.Noble if you delete it, all the work put into this answer would go to waste. I think it's better to keep it – Jackozee Hakkiuz Aug 06 '21 at 09:05