What if an automorphism fixes every maximal subgroup pointwise. Is it then the identity? [closed]

group-theory automorphism-group maximal-subgroup frattini-subgroup

This question came up in the discussion over here

My first thought was that then it fixes the Frattini subgroup. Any help?

For reference we found that the answer is no when each maximal subgroup is merely mapped back to itself.


The additive group of the rational numbers has no maximal subgroups, yet it admits many non-trivial automorphisms $x \mapsto kx$ for fixed $k \ne 0$.


Small counterexample: $G=C_4$, the cyclic group of order 4, say generated by $a$. The automorphism $a \rightarrow a^{-1}$, fixes the maximal subgroup $ \langle a^2 \rangle$ pointwise.

Related

When the denominator is larger than the numerator, why does the modulo equal the numerator?
Royden's proof that $L^{p_2}\subseteq L^{p_1}$ if $p_1<p_2$
Evaluate $\int_0^R e^{\frac{-r}{a}}r^2\mathrm dr$
Solve a simple equation involving limits for $f$, $\lim_{y\to x}\frac{(x-y)^2}{f(x)-f(y)}=1$
Inferring behavior of a function on the plane from curve integral
Eliminate $\alpha,\beta,\gamma$ from the system of equations
Showing that the space is normal.(exercise) [duplicate]
Transition matrix exercise
Find a unit normal vector at a point
Let $S_n$ be a simple symmetric random walk. Show that $P(S_1S_2...S_{2n} \neq 0) = P(S_{2n} = 0)$
Why does conditioning over a conditional probability give this equation (from Harvard's STAT 110 problem set)?
From a shell script, how can I check whether a table in MySQL database exists or not?