Do you have an idea how to tackle this problem:
The center of a finite group G whose order is 44, has an element of order 2.
I believe the idea must somehow involve the Sylow's Theorem; for example the Sylow 11-subgroup, say S, is normal. If we put C:=C(S) then C:=C(S)≤N(S)=G. We have that G/C is isomorphic to a subgroup of Aut(S), which has order φ(11)=10. Thus the order of G/C divides both 10 and 44; hence it could be either 1 or 2. No more progress yet!
In fact I had already asked a quite similar question, but the same method does not seem to work,,,
Thank you for considering this problem!