One way to ensure this happens is to have every maximal subgroup be characteristic. To get every maximal subgroup normal, it is a good idea to check p-groups first. To make sure the maximal subgroups are characteristic, it makes sense to make sure they are simply not isomorphic. To make sure there are not too many maximal subgroups, it makes sense to take p=2 and choose a rank 2 group.
In fact the quasi-dihedral groups have this property. Their three maximal subgroups are cyclic, dihedral, and quaternion, so each must be fixed by any automorphism.
So a specific example is QD16, the Sylow 2-subgroup of GL(2,3).
Another small example is 4×S3. It has three subgroups of index 2, a cyclic, a dihedral, and a 4 acting on a 3 with kernel 2. Since these are pairwise non-isomorphic, they are characteristic too. It also just so happens (not surprisingly, by looking in the quotient 2×2) that every element is contained in one of these maximal subgroups.
Here are my notes, since I like fairly explicit answers.
In case anyone wants to see the other examples more briefly:
Elementary abelian p-groups of order pn are found TI in PSL(2,pn). Such subgroups are uniquely identified by which subspace they stabilize.
Quaternion 2-groups are found TI in the their semi-direct products with their (unique) faithful irreducible module over a prime field of odd order. The key point is that the unique element of order 2 acts fixed-point-freely on the module, and so is not contained in the intersection of any two distinct Sylow 2-subgroups.
A weaker question has a positive answer:
For every p-group P is there a finite group G such that for some g in G, P ∩ Pg = 1?
Yes. For any p-group P, take G to be the semi-direct product of P with a large enough faithful module V over a finite field of characteristic not p. Then consider the union of CV(x) as x varies over P. Since V is faithful, each centralizer is a proper subspace, and if V has large enough dimension, it cannot be written as the union of |P| proper subspaces. If v is some element of V outside that union of centralizers, then P ∩ Pv = 1.
For p = 2, the classification is due to Suzuki (1964) who discovered his infinite family of finite simple groups in this same line of investigation. Not only did he classify the possibly P, but also the possible G. Let N be the largest odd-order normal subgroup of G.
- P is cyclic and G = P ⋉ N
- P is quaternion of order 8, and either G = P ⋉ N, or G / N ≅ SL(2,3)
- P is generalized quaternion and G = P ⋉ N
- P is elementary abelian of order 2n and PSL(2,2n) ≤ G / N ≤ PΓL(2,2n)
- P is the Sylow 2-subgroup of PSU(3,2n) and PSU(3,2n) ≤ G / N ≤ PΓU(3,2n) — P is sort of a GF(2n) version of the quaternion group of order 8.
- P is the Sylow 2-subgroup of Sz(2n) and G / N = Sz(2n)
In other words, I missed two infinite families (and one fusion system). The structure of N is restricted, but not I think not classified. In some sense, one should read this list as "there exists an N such that...".
The classification for odd p appears to be a post-CFSG result, and is closely related to strongly embedded subgroups (similar to the concept Arturo mentions below: malnormal). The strongly embedded list is on page 383 - 384 of number 3 of the GLS writeup of the CFSG. The TI list is in Blau-Michler (1990), but is based on older lists I have not yet tracked down.
Other than cyclic and elementary abelian, there are a few more infinitely families, and a few sporadic exceptions.
Bibliography:
Suzuki, Michio.
"Finite groups of even order in which Sylow 2-groups are independent."
Ann. of Math. (2) 80 (1964) 58–77.
MR162841
DOI:10.2307/1970491
Blau, H. I.; Michler, G. O.
"Modular representation theory of finite groups with T.I. Sylow p-subgroups."
Trans. Amer. Math. Soc. 319 (1990), no. 2, 417–468.
MR957081
DOI:10.2307/2001249
Best Answer
You say that $G$ cannot be finite. But what about $G = \mathbb Z / 2 \mathbb Z \oplus \mathbb Z / 2 \mathbb Z$?