Recall that a group G is said:

1) *Sylow $\pi$-connected* if the Sylow $\pi$-subgroups of G are conjugates in G.

2) *Sylow $\pi$-integrated* if every subgroup of G is Sylow $\pi$-connected.

3) *Completely Sylow integrated* if G is Sylow $\pi$-integrated for every set of primes $\pi$.

Now i found in some work (for example in Dixon or Hartley) that:

1) *''For by a well known theorem of P. Hall, a finite completely Sylow
integrated group is soluble...''*

2) *''...and so an arbitrary completely Sylow integrated group is locally soluble''.*

I don't found nowhere a proof for 1); for 2) i must consider a finitely generated (and no necessarily finite) subgroup of a completely Sylow integrated group for proof the locally solubility? (pheraps G is assumed locally finite?)

Help me, please.

$\pi$-elementif $|\langle g\rangle|$ is finite and if prime divisors of $|\langle g\rangle|$ are in $\pi$. 2) G is said a$\pi$-groupif every element of G is a $\pi$-element. 3) ASylow $\pi$-subgroupof G is a maximal element in the ordered set ({P$\le$ G: P is a $\pi$-group},$\subseteq$) $\endgroup$