# CS-Baer Lie group

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## Definition

A CS-Baer Lie group is a group $G$ satisfying both the following two conditions:

1. $G$ is a group of nilpotency class two, i.e., its nilpotency class is at most two. Equivalently, the derived subgroup $[G,G]$ is contained in the center $Z(G)$ of $G$.
2. There exists a subgroup $H$ of $G$ such that $[G,G] \le H \le Z(G)$, and every element of $[G,G]$ has a unique square root within $H$.

## Examples

### Finite examples

For finite groups, being a CS-Baer Lie group is equivalent to being a LCS-Baer Lie group, which in turn means that it is a group of class two whose 2-Sylow subgroup is abelian.

### Infinite examples

There are examples of infinite groups that are CS-Baer but not LCS-Baer. The smallest example is central product of UT(3,Z) and Z identifying center with 2Z. Note that this particular example is a LUCS-Baer Lie group (see LUCS-Baer Lie group#Examples).

We can create examples of CS-Baer Lie groups that are neither LUCS-Baer Lie groups nor LCS-Baer Lie groups, by combining features of the above examples. Specifically, let $G$ be the group central product of UT(3,Z) and Z identifying center with 2Z. Then, the group $G \times \mathbb{Z}/2\mathbb{Z}$ is a CS-Baer Lie group but it fails to be a LUCS-Baer Lie group (since it has 2-torsion within the center). It also fails to be a LCS-Baer Lie group (since the derived subgroup itself does not allow for halving).

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions