Name for a certain class of groups that contains all the abelian groups I cam across this type of groups. Is there a name for groups that satisfy this condition:
$$\forall x,y\in G[xyx^{-1}\in \langle y\rangle]$$
As mentioned in the title, it is easy to see that all the abelian groups satisfy this condition.
Thank you
 A: If $G$ is a group such that for all $x,y \in G$, $xyx^{-1} \in \langle y \rangle$, then every cyclic subgroup is normal. Conversely if every cyclic subgroup is normal, then $G$ satisfies the condition.
More commonly this is expressed as every subgroup is normal: if every cyclic subgroup is normal, and $H \leq G$ is any subgroup, then $H$ is the product of the cyclic subgroups $\langle y \rangle$ for $y \in H$, so $H$ too is normal. The converse is clear.
A group in which every subgroup is normal is called a Dedekind group. A non-abelian Dedekind group is of the form $Q_8 \times P \times V$ where $P$ is a periodic abelian group with no elements of even order and $V$ is an abelian group all of whose elements have order dividing 2. These were classified by Dedekind (1897) in the finite case and Baer (1933) in general. A modern proof is theorem 5.3.7 on page 139 of Robinson (1982).


*

*Dedekind, R.; 
“Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind.”
Math. Ann. 48 (1897), no. 4, 548–561.
MR1510943
JFM28.0129.03
DOI:10.1007/BF01447922

*Baer, R.
“Situation der Untergruppen und Struktur der Gruppe.”
[J] Sitzungsberichte Heidelberg (1933), Nr. 2, 12-17 (1933).
JFM59.0143.02

*Robinson, Derek J. S.
“A course in the theory of groups.”
Graduate Texts in Mathematics, 80.
Springer-Verlag, New York, 1982. xviii+481 pp.
ISBN: 0-387-90600-2
MR648604
DOI:10.1007/978-1-4684-0128-8
DOI:10.1007/978-1-4419-8594-1
