Examples of Sylow $p$-subgroups in infinite groups Can you provide some examples of Sylow $p$-subgroups in infinite groups? 
(Sylow $p$-subgroups are $p$-subgroups that are maximal with respect to inclusion among all $p$-subgroups in the group.)
 A: Here's a very simple example of a group which has two maximal $p$-subgroups which are not conjugate. In fact, it's a special case of the comment I made. Let $G = \langle a,b : a^2 = abab = 1 \rangle.$ Then $G$ is an infinite dihedral group with an infinite cyclic normal subgroup $B = \langle b \rangle.$ it is easy to check that every element of the coset $aB$ has order $2,$ whereas all non-identity elements of $B$ have infinite order. Hence all maximal $2$-subgroups of $G$ have order $2.$ We claim that $a$ and $ab$ are not conjugate in $G.$ For, given any integer $j,$ we have $(ab^j)^{-1}a(ab^j) = b^{-j}ab^j = aab^{-j}ab^j = ab^{j}b^{j} = ab^{2j}.$ Hence we see that all elements of the form $ab^{2j}$ are conjugate to $a,$ and similarly (or just allowing $ab$ to play the role of $a),$ we see that all elements of the form $ab^{2j+1}$ are conjugate to $ab.$
A: I'll give you some examples. 
You can consider $\bigoplus_{n=0}^\infty \mathbb Z/p\mathbb Z$, it is a $p$-Sylow by itself, or if you prefer something which isn't a $p$-group by itselfs can consider the group $\mathbb Z \oplus \bigoplus_{n=0}^\infty \mathbb Z/p\mathbb Z$
other. 
Another example could be $\mathbb Z \oplus \mathbb Z/p\mathbb Z$, in this case the $p$-Sylow is finite.
As you can see you can create as many example as you wish. 
A: Maximal $p$-subgroups can become quite ugly in infinite groups, even if the groups are locally finite (i.e., each finitely generated subgroup is finite).
Take for example a look at the construction in:
Marianna Dalle Molle, Sylow Subgroups Which Are Maximal in
the Universal Locally Finite Group of Philip Hall, Journal
of Algebra 215, 229-34 (1999)
