Understanding the Automatic Formation of Lie Subgroups in a Lie Group

[hunhengy]

Recently, I read the follow paragraph:

Let $G$ be a Lie group. If $H$ is a subgroup defined by the vanishing of a number of (continuous) real-valued functions
$$H=\{g\in G| F_i(g)=0, i=1,2,\cdots,n\},$$
then $H$ is automatically a Lie subgroup of $G$. We do not need to check the maximal rank conditions of the $F_i$.

Why is $H$ autonmatically a Lie subgroup of $G$? Why should not need to check the maximal rank conditions of the $F_i$?

 

[arkajad]

This statement is not true as it is easy to construct a couter example, for instance for G=R.The statement would be true if you would require f(ab)=f(a)+f(b), f(e)=0.

 

[eok20]

The statement is true (since we're assuming H is a subgroup).

Are you familiar with the theorem that a (topologically) closed subgroup of a Lie group is a Lie group?

 

[arkajad]

You are right. I missed the part that H was assumed to be a subgroup.

 

[hunhengy]

I know this statement is right. The only thing is that I can not check it by the definition of Lie Group? Why should not need to check the maximal rank conditions of the $F_i$?