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$\begingroup$

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

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    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 – 2017-02-24
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    I think,group itself is non-trivial subgroup. And if H – 2017-02-24
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    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. – 2017-02-24
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    @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](https://en.wikipedia.org/wiki/Trivial_group). – 2017-02-24
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    oh thanks for clearing that up. My professor always used trivial this way so it stuck with me. – 2017-02-24
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    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. – 2017-02-24

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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 $\{e\}$.