Classify the groups of order $88$ up to isomorphism.

Classify the groups of order $88$ up to isomorphism.

Here is what I have so far (I'm aware that there are $12$ groups, but I don't know which ones I'm missing as well as why the $3$ groups are abelian and the other $9$ are non-abelian.)

We are asked to classify the groups of order $88$ up to isomorphism. To prove that any group of order $88$ is abelian. So if $|G|=88$, then we can construct all abelian groups of order $88$ by using the Fundamental Theorem of Finitely Generated Abelian groups. We can use the Fundamental Theorem of Finitely Generated Abelian Groups wich states the following:

Let $G$ is a finitely generated abelian group. Then
\begin{equation}G\cong\Bbb Z^r\times\Bbb Z_{n_1}\times \Bbb Z_{n_2}\times\dots\times\Bbb Z_{n_s},\tag1\end{equation} For some integers $r,n_1,n_2,\dots,n_s$ satisfying the following conditions:

1. $r\ge 0$ and $n_j\ge 2$ for all $j$, and
2. $n_{i+1}\mid n_i$ for $1\le i\le s-1$

The expression in $(1)$ is unique: if $G\cong\Bbb Z^t\times\Bbb Z_{m_1}\times \Bbb Z_{m_2}\times\dots\times\Bbb Z_{m_u}$, where $t$ and $m_1,m_2,\dots,m_u$ satisfy 1. and 2. (i.e., $t\ge 0$,$m_j\ge 2$ for all $j$ and $m_{i+1}\mid m_i$ for $1\le i\le u-1$), then $t=r$,$u=s$ and $m_i=n_i$ for all $i$.

This gives us an effective way of listing all finite abelian groups of a given order. Namely, to find (up to isomorphism) all abelian groups of a given order $n$ one must find all finite sequences of integers $n_1, n_2,\dots,n_s$ such that

1. $n_j\ge 2$ for all $j\in\{1,2,\dots,s\}$,
2. $n_{i+1}\mid n_i$,$1\le i\le s-1$, and
3. $n_1 n_2\dots n_s=n$

We can also note that every prime divisior of $n$ must divide the first invariant factor $n_1$. In particular, if $n$ is the product of distinct primes, which are all to the first power, which is called squarefree, we see that $n|n_1$, hence $n=n_1$. This proves that if $n$ is squarefree, there is only one possible list of invariant factors for an abelian group of order $n$. The factorization of $n$ into prime powers is the first step in determining all possible lists of invariant factors for abelian groups of order $n$.

This means that we can break 8$8$ down into its prime factors which would give us the following:$$88=2\cdot 44=2\cdot 2\cdot 22=2\cdot 2\cdot 2\cdot 11$$So if we say that $n=88=2^3\cdot 11$, as we have stated above we must have that $2×11|n_1$, so possible values of $n_1$ are as follows:$$n_1=2^3\cdot 11~\lor~n_1=2^2\cdot 11~\lor~n_1=2\cdot 11$$For each of these we need to work out the possible $n_2$’s. For each resulting pair $n_1,n_2$ we need to then work out the possible $n_3$’s and then continue in this manner until all lists satisfying 1. and 3. are obtained.Therefore $88$ can be written as $2^3\cdot 11$. Which would give us the following:

Order $p^\beta$: Partitions of $\beta$ Abelian Groups$$2^3:~3,~\Bbb Z_8;~~~2,1,~\Bbb Z_4\times\Bbb Z_2;~~~1,1,1,~\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_2$$$$11^1:~1,~\Bbb Z_{11}$$

We can obtain the abelian groups of order $88$ by taking one abelian group from each of the two lists above and taking their direct product. Doing this in all possible ways gives all isomorphism types:$$\Bbb Z_{88},~\Bbb Z_8\times\Bbb Z_{11},~\Bbb Z_4\times\Bbb Z_2\times\Bbb Z_{11},~\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_{11},~\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_{22},~\Bbb Z_4\times\Bbb Z_{22},~\Bbb Z_2\times\Bbb Z_{44}$$When we have completed this we will have $12$ groups. By the Fundamental Theorems above, this is a complete list of all abelian groups of order $88$, every abelian group of this order is isomorphic to precisely one of the groups above and no two of the groups in this list are isomorphic.

We can then define abelian and non-abelian groups. Abelian groups or commutative groups are groups in which the results of applying the group operation to two group elements does not depend on the order in which they are written, in other words these groups are are groups that following the axiom of commutativity.

Abelian groups generalize the arithmetic of addition of integers. Non-abelian groups, also known as non-commutative groups are groups $(G,*)$ in which there exists at least one pair of elements $a$ and $b$ of $G$, such that $a*b\ne b*a$.

Of these $12$ groups $3$ of them are abelian and the other $9$ are non-abelian groups. The three abelian groups are $\Bbb Z_{88}$, $\Bbb Z_4\times\Bbb Z_{22}$, and $\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_{22}$.