Is the Relation R on Groups an Equivalence Relation?

[gbean]

Homework Statement

The relation R on the set of all groups defined by HRK if and only if H is a subgroup of K is an equivalence relation.

Homework Equations

Subgroup: has identity, closed under * binary relation, has inverse for each element.
Equivalence relation: transitive, symmetric, and reflexive.

The Attempt at a Solution

I know that the answer is false, but I'm having trouble parsing the question. Any help would be greatly appreciated!

 

[CompuChip]

The statement is about the equivalence relation. So if you want to prove that R is or is not an equivalence relation, you will need to check the three properties of an equivalence relation.

Can you express more concretely what the three properties say in this case (say, if H, J, K are arbitrary groups).

 

[gbean]

The statement is about the equivalence relation. So if you want to prove that R is or is not an equivalence relation, you will need to check the three properties of an equivalence relation.

Can you express more concretely what the three properties say in this case (say, if H, J, K are arbitrary groups).

So is this still the same question? (I reworded it a bit to make it more understandable for myself): The relation R is an equivalence relation on the set of all groups defined by HRK if and only if H is a subgroup of K.

So R is not necessarily an equivalence relation because 3 conditions were not satisfied:
HRH for all groups, so H is a subgroup of itself, which is true, so reflexive is satisfied.
If HRK, then KRH. If H is a subgroup of K, then K is a subgroup of H. This is false, so symmetry does not follow.
Assume HRK and KRJ, then HRJ. If H is a subgroup of K, and K is a subgroup of J, then H is a subgroup of J. This is true, so transitivity is satisfied.

 

[CompuChip]

Exactly.
Note that the "if and only if" part is in the definition of R, it does not apply to it being an equivalence relation. That means:

R is defined by the following statement: HRK is true if and only if H is a subgroup of K​

which is something else than

R is an equivalence relation, if and only if it is true that H is a subgroup of K​

which is clearly nonsense (you have not even said what H and K are there).