$\frac{3}{2}$-generated groups?

Question

Is there a special name for groups $G$ with the following property?

For every $g \in G \setminus \{1\}$ there is some $h \in G \setminus \langle g \rangle$ with $G = \langle g,h \rangle$.

Which symmetric groups have this property? (I have already checked with a program that $S_3,S_5,S_6,S_7,S_8,S_9$ have this property. $S_4$ does not have this property.)

Edit. $S_n$ has this property for all $ n \neq 2$ (see the accepted answer). Is there a proof for this in English?

Answer 1

Such groups are called $\frac{3}{2}$-generated. According to Breuer, Guralnick and Kantor, a finite group is conjectured to be $\frac{3}{2}$-generated, if and only if every proper quotient is cyclic. For the symmetric group $S_n$ this is true if and only if $n=4$, see this question. Hence we obtain:

Conjecture: The symmetric group $S_n$ for $n\neq 4$ is $\frac{3}{2}$-generated.

This was proved for $S_n$, $n\neq 4$ by G. Y. Binder in $1968$. A proof in English is available by I. M. Isaacs and Thilo Zieschang in Generating Symmetric Groups, $1995$.