Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid  ?

More precisely, I'm looking for classes of groups $\mathcal{C}$ with the following properties:

• $\mathcal{C}$ includes the finite groups
• in $\mathcal{C}$ there is a notion of Sylow subgroups that coincides with the usual one when restricted to finite groups
• Sylow's theorems (or part of them) are valid in $\mathcal{C}$

An example of such a class $\mathcal{C}$ is given by the class of profinite groups.

Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?

More precisely, I'm looking for classes of groups $\mathcal{C}$ with the following properties:

• $\mathcal{C}$ includes the finite groups
• in $\mathcal{C}$ there is a notion of Sylow subgroups that coincides with the usual one when restricted to finite groups
• Sylow's theorems (or part of them) are valid in $\mathcal{C}$

An example of such a class $\mathcal{C}$ is given by the class of profinite groups.