Find all the groups $G$ such that $|G|\leq 6$

Question

Problem statement:

Find all the groups of order at most 6.

Attempt at a solution:

What I thought was, if $|G|=1$, then the only possible element of the group is the neutral element. Now note that if $g \in G$, then $g^{-1}$ has to be in $G$ by definition of a group; by this observation if $|G|\leq 6$, then $G$ must be of the form $G=\{e,g,g^{-1}\}$ or $\{e,g,g^{-1},h,h^{-1}\}$, so $|G|$ can be $1$, $3$ or $5$.

Well, my doubt on the exercise is: is this what the problem is asking? I mean, do I have to explicitly give all the sets G that satisfy: $G$ is a group and $|G|\leq 6$ or is my answer correct? What I mean explicitly is, for example, say one of the groups is $S_3$, another one $S_2$.

I am not so sure my answer is correct because, it could happen that an element $g \in G$ is its own inverse and I am taking that case into account, but I really have no idea how to find explicitly all groups of order at most $6$. Also, I've noticed that $S_3$ and $S_2$ are clearly groups that satisfy the condition required but are not consider in my answer since $|S_2|=4$ and $|S_3|=6$,

Answer 1

It's not the name of the elements that matters, only the relationship between them. You missed the case in which an element is its own inverse, that is why some groups that you know of are not of the form you mention.

For example, the "group of two elements each of which is its own inverse" is an explicit enough characterisation of $\mathbb Z_2$. This characterises all groups with that property, which are isomporphic, so you can say that they all are $\mathbb Z_2$ "up to isomorphism". What they are asking is to find all groups of cardinality less than 6 up to isomorphism. You'll want to search for various examples and also think up some of them on your own, maybe search for subgroups of some known groups, or ask yourself which possibilities are left. Maybe doing an operation table can help you find some new combinations.

Remember that names don't matter: if you construct a group whose elements can be renamed so as to be equal to, say $S_2$, then you just found another version of $S_2$, and you shouldn't count it as another group.

Answer 2

Most of the time when such questions are asked, then its implicit in the question that you need to find all the group up to isomorphism. So if you need to find all the groups whose orders are $\leq 6$ then consider $5$. First of all can you find a group of order $5$? If yes, then suppose you have another group of order $5$, then are these two group isomorphic or are different? Similar things should be asked for all other group orders.