What are the groups classes that have some properties similar to abelian groups?

Question

Question : What are the groups classes that have some properties similar to abelian groups?

I want class of groups which follows the almost all the properties of abelian groups. In short I was looking for nilpotent and solvable groups.

Answer 1

I think this is vague, but here's a way to precise this up a bit.

You might call a group $G$ close to being abelian if it has an abelian subgroup $A$ of small index. The terms "close" and "small" are relative and flabby and can be made precise if you want.

For instance, say $G$ is close to being abelian if it has an abelian subgroup $A$ of index 2. That's probably the best possible definition without allowing $G$ itself to be abelian. In this case, you could get such $G$'s by dihedral extensions. Take any abelian group $A$, and let the group of order 2 act on it by inversion. This gives a group $D(A)$ of order twice that of $A$, containing $A$ as an index-2 subgroup. (When $A$ is cyclic of order $n$, this $D(A)$ is the ordinary dihedral group of order $2n$). Hence $D(A)$ is close to being abelian. As long as not all elements of $A$ square to the identity, this group $D(A)$ will be non-abelian but "close."

YMMV, take this for what it's worth.

Answer 2

Soluble groups (solvable in US English) and nilpotent groups are generally considered to be generlisations of abelian groups. Indeed, abelain $\subset$ nilpotent $\subset$ soluble. Note that homomorphic images of abelian groups are also abelian. This is also true for soluble and nilpotent groups: homomorphic images of soluble/nilpotent groups are also soluble/nilpotent (delete as appropriate).

Define a commutator as $[h, k]:=h^{-1}k^{-1}hk$ for $h, k\in G$, and for $H, K\leq G$ define $[H, K]:=\langle [h, k]: h\in H, k\in K\rangle$. Note that if $G$ is abelain then $[h, k]=1$ for all $h, k\in G$.

Soluble groups. Define $G^{(0)}=G$, and $G^{(i+1)}=[G^{(i)},G^{(i)}]$. A group is soluble if there exists some $n\in\mathbb{N}$ such that $G^{(n)}=\{1\}$. For example, a group is abelian if and only if $G^{(1)}=\{1\}$. For non-examples, every non-abelian simple group is not soluble: $G^{(1)}\unlhd G$ so $G^{(1)}$ is either $G$ or trivial, but by assumption $G^{(1)}\neq1$. Hence, $G=G^{(0)}=G^{(1)}=G^{(2)}=\ldots$. For a major theorem, the Feit-Thompson theorem states that every group of odd order is soluble. Hence, every non-abelian simple group has even order. For more information on soluble groups, see wikipedia.

Nilpotent groups. Define $\gamma_1(G)=G$, and $\gamma_{i+1}(G)=[\gamma_{i}(G), G]$. A group is nilpotent if there exists some $n\in\mathbb{N}$ such that $\gamma_{n}(G)=\{1\}$. For example, a group is abelian if and only if $\gamma_{1}(G)=\{1\}$. Also, every group of prime-power order is nilpotent. For a non-example, the group $S_6$ is soluble but not nilpotent. For more information on nilpotent groups, see wikipedia.

The difference between nilpotent groups and soluble groups is that solubility tests the commutativity of the groups $G^{(i)}$, while nilpotency tests how much the groups $\gamma_i(G)$ commute with the whole group $G$.