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I know that subgroup of index $2$ is normal.

I am interested in knowing that is that subgroup is unique or there exist example of subgroup which can have two subgroup of index $2$?

If there is example exist , then what is condition on subgroup implies that subgroup of index $2$ is unique?

Any Help will be appreciated.

Unfortunately it is not even unique up to isomorphism! Take, for instance, $(mathbbZ/2mathbbZ)times (mathbbZ/4mathbbZ)$.

No way is it unique (generally). The group $G=mathbbZ_2 times mathbbZ_2$ has three subgroups of index $2$: $mathbbZ_2 times0$, the vice versa, and $(0,0), (1,1)$. You can use the same idea to cook up plenty of other examples.

It is maybe worth noticing the following. Let $G$ be a group and let $I_2(G)=#=2$, be the number of subgroups of index $2$.

Theorem (Crawford, Wallace, 1975) Let $G$ be a group and $n$ a non-negative integer. Then $I_2(G)=n$ if and only if $n=2^k-1$ for some non-negative integer $k$.

See On the Number of Subgroups of Index Two-An Application of Goursat's Theorem for Groups, R. R. Crawford and K. D. Wallace, Mathematics Magazine Vol. 48, No. 3 (May, 1975), pp. 172-174.

From this it follows that $I_2(G) equiv 1$ or $3$ mod $6$, and in particular, $I_2(G) neq 2$.

Another observation (see also the quoted paper): the following are equivalent.

(a) A group $G$ has a unique subgroup of index $2$.

(b) $G$ cannot be expressed as the union of 3 different subgroups.

(c) $G$ does not have a quotient isomorphic to Klein's group $V_4$.

You can actually make arbitrarily large groups where every element has order 2: Given $X$, the power set of $X$, $mathcalP(X)$, is a group under the operation $Delta$, known as symmetric difference ie for $A, B subseteq X$, $A Delta B = (A cup B) - (A cap B)$, and you further have that $emptyset$ is the identity, and $A Delta A = emptyset$.

Even further, this class of groups are all commutative, so all these two element subgroups are all normal.