Questions about Sylow $p$-groups

Question

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group?

Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group then so is $gHg^{-1}$, hence $H=gHg^{-1}$ by uniquness?

Question 3 Can someone please give me an example of a Sylow $p$-group in some group which is not normal?

Answer 1

Question $1$ is true, if $P$ and $Q$ are $p$-sylow subgroups then they are conjugate. In an abelian group every subgroup is normal.

Question $2$ is true, if you conjugate a $P$-sylow subgroup you get another (because they have the same order), so if there is only one then the subgroup is invariant under conjugation, hence normal.

Question $3$: any group in which there is more than one Sylow $p$-subgroup, For example $S_3$ and $p=2$.

Answer 2

1. Yes: in an abelian group every subgroup is normal.

2. Correct.

3. The group $A_5$ is simple and it has $60$ elements. Then $\langle(1,2,3,4)\rangle$ is a Sylow $2$-group and it is not normal.