# What examples of groups should students in abstract algebra learn to test ideas on?

Having seen many people go through abstract algebra courses, I've noticed that one factor for success is the ability to pull out examples to test ideas on (e.g. if two subgroups have the same index, are they the same? No, the Klein four-group is a quick counterexample).

Now, my example may seem trivial, but that's the point; many students who struggle with abstract algebra don't see such things as trivial, and try to prove or disprove using symbolic manipulations instead of quickly going through a list of examples.

What are the core examples of groups that students should think of when testing ideas?

Please indicate the groups and why you think they are important.emphasized text

• I think this question is answered pretty well with en.wikipedia.org/wiki/List_of_small_groups But I leave you with the question: Suppose finite groups $G, H$ have a bijection $b: G \rightarrow H$ such that for all $g \in G$, we have $ord(g) = ord(b(g))$. Must $G$ and $H$ be isomorphic? (If yes: prove it. If no: what is the smallest counterexample?) Apr 1 '14 at 23:47
• The list I posted at Groups of real numbers could be of use. However, I used those groups mainly for practice in verifying group axioms for strange looking groups, whereas your interest is a little different. Apr 7 '14 at 20:06
• John Jones made this fantastic tool. I use it extensively when I teach abstract algebra: hobbes.la.asu.edu/groups/groups.html Feb 26 '21 at 15:41

$\mathbb Z_n$ since it is very easy to compute in and you have one of order $n$ for every $n\ge 1$.

$S_3$, since it's the smallest non-abelian group.

$S_4$, since $S_3$ is sometimes too small.

$A_4$, Since it is the smallest group that does not contain a subgroup of every possible order dividing its order.

$\mathbb Z_2 \times \mathbb Z_2$ since it's the smallest non-cyclic abelian group.

$\mathbb Z$ since it's infinite.

$\mathbb R$ since it's infinite but not cyclic.

$\mathbb Q$ since it's infinite and countable, but still not cyclic.

$S_n$ due to their immense importance and their complexity.

$A_n$ since they have index two in $S_n$ (for $n>1$).

$D_n$ for their geometric content.

I'm sure I'm missing some more crucial groups...

• The generalized quaternion groups (especially the usual one) are also important for counter examples at some point (non-abelian with all subgroups normal, $p$-group with unique subgroup of order $p$ but not cyclic). Also, the group of $3\times 3$ upper triangular unipotent matrices over a field with $p$ elements (when $p\geq 3$ this is non-abelian with all non-identity elements of the same order). And I suppose the Tarski monsters are also good to know just to ensure that all your intuition about what is possible is wrong. Apr 1 '14 at 9:15
• I like A5 since their conjugacy classes split. Apr 1 '14 at 9:54
• Actually, $S_4$ does satisfy the converse to Lagrange's theorem. The alternating group $A_4$ has no subgroup of order $6$, and this is the smallest example. Also, linear groups ($GL_n(F)$, $SL_n(F)$ etc) are extremely important. Apr 1 '14 at 17:23
• @MikkoKorhonen thanks!! (corrected). Apr 1 '14 at 21:02
• For much the same reason as you list $S_4$, I remember that working with $(\Bbb Z_2)^n$ was sometimes helpful (though I can't provide any good examples for this)
– user37
Apr 3 '14 at 7:29

Student should know some groups coming from geometry, which are often overlooked (sure, $A_4$, $S_n$, $A_5$, $D_n$ and others previously cited do arise in geometry...)

In my opinion the most important examples are the affine groups $\mathrm{Aff}(\mathbb{R}^n)$ (in fact, already $\mathrm{Aff}(\mathbb{R})$ is great): it helps remember how semidirect products work (which subgroup acts on which one? Linear maps act on translations; which one is normal? Affine maps have a linear part so the set of translations is the kernel of a map, hence is normal). I consider I really understood skew product when I understood $\mathrm{Aff}(\mathbb{R})$.

• +1; The skew product is often called the semidirect product.
– kan
Apr 3 '14 at 3:57
• @kan thanks, I use this term mostly in teaching (in french), so my english is a bit weird sometimes. I'll edit. Apr 3 '14 at 9:30