Uniqueness of a normal subgroup

Question

If I know that a subgroup of a finite group is a normal subgroup, is this condition sufficient to state that it is unique? I think that is sufficient thanks to Sylow's theorems, but in any exercises I've found that I must prove it is unique also if I already know it is normal.

Answer 1

The answer is no, there might be other subgroups of the same order. As an easy example, look at $\mathbb{Z_2}\times\mathbb{Z_2}$. It has $3$ distinct subgroups of order $2$, obviously all are normal. (since the group is abelian)

Now, specifically for Sylow subgroups it is true that a Sylow subgroup of a specific order is unique if and only if it is normal. It indeed follows from the second Sylow theorem which says that each two $p$-Sylow subgroups of a group $G$ are conjugate to each other.

Answer 2

No. An example would be $G=C_2 \times A_4$. The first factor is a normal subgroup isomorphic to $C_2$, but $G$ has other subgroups of order $2$ (inside $A_4$) that are not normal.

Answer 3

If a subgroup $H$ has an order that is unique, then $H$ must be normal, even characteristic.