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When studying abstract algebra, I do prefer having a nice simple and concrete example to demonstrate the theorem/lemma.

However, the 'first course' book that I am currently learning from often uses (in my opinion) quite complicated examples to demonstrate the theorem/lemma. Sometimes this is fine, but other times I may as well be reading Chinese.

For example, my 'go-to' group for an infinite abelian group is $\left ( \mathbb{Z}, + \right )$. My 'go-to' group for a finite abelian group is $\left ( \mathbb{Z_n}, + \right )$

What are some nice and easy-to-understand

  1. Non-abelian finite groups
  2. Non-abelian infinite groups

that I could use to demonstrate/test the theorems I learn?

I understand that 'easy' is a subjective term. But I am trusting your expertise as to what would be considered 'easy' for someone first learning abstract algebra.

Trogdor
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6 Answers6

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$S_3$ is quite easy to understand and non-abelian, but you should spend some time getting used to it.

It has six elements: the identity, the three transpositions each swapping a pair of elements, and the two "rotations", which rotate the three elements in one direction or in the other.

It's clear what the square of a transposition is, and what the product of two rotations is, but it's less obvious what is the product of a transposition and a cycle, or of two distinct transpositions. Get yourself three distinct objects and play around with them until you understand how it works and can explain intuitively why the multiplication table for $S_3$ is what it is.

Then work out the subgroups, and which of those subgroups are normal.

Jack M
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Semidirect products also give very concrete ways of understanding nonabelian groups. For example, I've always found that the semidirect product decomposition of diheldral groups $D_{2n}$ as $\mathbb{Z}/n\mathbb{Z}\rtimes\mu_2$, where $\mu_2 = \{\pm 1\}$ always made it much easier to do computations with them.

In this case, it tells you that every element of $D_{2n}$ can be written as $(a,\pm)$, where $a\in\mathbb{Z}/n\mathbb{Z}$.

Multiplication is given by $(a,-)(b,\pm) = (a-b,\mp)$, and $(a,+)(b,\pm) = (a+b,\pm)$.

Inversion is given by $(a,\pm)^{-1} = (\mp a,\pm)$.

oxeimon
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I would advise using groups that can be understood in a geometrical way, and not only by their presentation and abstract definitions. Therefore, groups of symmetries are always good examples.

You could use: $\mathbb{Z}_2$ seen as reflection, the circle $S^1$ seen as a group of rotations, the dihedral groups seen as symmetries of polygons, the symmetric groups $S_n$ as groups of automorphisms of finite sets, and if they have seen a bit of physics you have $SO(3)$ as rotational symmetries and $SO(3,1)$ (proper Lorentz transformations). You could also take a look at lattice groups and so on. All of these are interesting examples because they arise from other concrete problems, and not only as an abstract mathematical construction.

Daniel Robert-Nicoud
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Non-Abelian finite: $D_{2n}$

Non-Abelian Infinite: $GL_2(\mathbb{R})$

AnotherPerson
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A couple of non-Abelian infinite groups:

  • the multiplicative group of invertible 2x2 matrices over $\mathbb{Q}$
  • the free group on 2 generators.

Both are countable.

BrianO
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In addition to the answers already posted, don't forget that the direct product of two groups is a group, and that lets you put together many more examples.

Also, I don't think anyone mentioned the quaternion group.

Gerry Myerson
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