# Does there exist two trivial subgroups?

I'm so confused. I know the trivial subgroup is identity itself.

• Hi when people mean trivial subgroup, they mean the group itself(which is a subgroup) and the identity element(also a subgroup).so yes there are two trivial subgroups – asddf Feb 24 '17 at 20:07
• I think,group itself is non-trivial subgroup. And if H<G then it is called proper subgroup of G.(book reference contemprory Abstract algebra By Galliaon – Nimesh Feb 24 '17 at 20:08
• Keep in mind that trivial here is used in the meaning of very easy. When learning the definition of a subgroup its very easy to see that the identity element and the whole group are subgroups. Thats why they are called trivial. You didnt mention proper subgroups in your post, you just said subgroups.If you want to go for proper subgroups only, you are right only e is a proper trivial subgroup. – asddf Feb 24 '17 at 20:18
• @asddf You are using trivial in the informal way commonly seen in mathematics, but in group theory, it refers particularly to the group with one element. See, for example, Wikipedia. – Théophile Feb 24 '17 at 20:26
• Given the inconsistent usage around, I think $-4$ is a bit harsh for this question! I wouldn't normally upvote it, but on this occasion I will. – Derek Holt Feb 24 '17 at 21:20

## 1 Answer

A trivial group is a group with only one element: $\{e\}$. Since all such groups are isomorphic, we often speak of the trivial group.

Every group clearly has the trivial group as a subgroup. A proper subgroup of a group $G$ is a subgroup that is not $G$ itself. If a group $G$ has no proper nontrivial subgroups, then its only subgroups are $G$ and $\{e\}$.