Difference between conjugacy classes and subgroups?
I am studying Group theory and Im not sure I understand the difference between a conjugacy class and subgroup?
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$\begingroup$Conjugacy classes are the orbits of elements of the group, under the action of conjugation. It has nothing to see with subgroups.
For instance, in an abelian group, the conjugacy classes are simply the singletons made up of the elements of the group, while a subgroup usually has more than one element…
$\endgroup$ $\begingroup$A conjugacy class is a set of congugated elements, it is not necessarily a subgroup (to start with, the identity $e$ is not part of any conjugacy class, except $\{e\}$). Normal subgroups are a union of conjugacy classes, but the converse is not true. The cardinality of a conjugacy class is counted by the index of a subgroup: $|Cl_G(g)|=|G:C_G(g)|$, where $C_G(g)$ is the centralizer of $g$, the subgroup of all elements of $G$ that commute with $g$. The singletons among the conjugacy classes are hence the elements that are in the center $Z(G)$ of $G$.
$\endgroup$ $\begingroup$As others said subgroup has all the properties of Group. But conjugacy classes are just the set, but created with conjugacy and are equivalence relation.
- Intuitively conjugacy is, looking the same thing with different perspective. For ex. take ${D_6}$ , a hexagon and say r=clockwise rotation and f=horizontal reflection. Now consider ${rsr^{-1}}$ , rather than rotating hexagon, we shift our place to one above (anticlockwise rotation) and then apply ${s}$ i.e., horizontal reflection from the axis our point of view. and then undo our rotation and come back to our original place.
- Now s is still a reflection , but viewed in different perspective. Here ${s}$ is conjugated by ${r}$ and ${r}$ determines how you will change your perspective.
Now, if we see Subgroup this way, then it is a set H, where composing symmetries in H remains in H (i.e, yield one of the element of H), undoing any symmetry will remain in H. Obviously, doing nothing (Identity) is also in H.
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