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?

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    $\begingroup$ Why not $S_4$?? $\endgroup$ – Dietrich Burde May 3 '17 at 11:27
  • $\begingroup$ In $S_4$ it doesn't work for $g=(1,2)(3,4)$. $\endgroup$ – Derek Holt May 3 '17 at 12:18
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    $\begingroup$ It is known that all finite simple groups have this property: see the discussion here Apparently this property is called 3/2-generated. $\endgroup$ – Derek Holt May 3 '17 at 12:20
  • $\begingroup$ Thank you Derek! Unfortunately my google skills do not help me out here: How to get information about 3/2-generated groups? (In particular, concerning the symmetric groups.) $\endgroup$ – HeinrichD May 3 '17 at 13:30
  • $\begingroup$ I found some links by googling. See for example here and the references given. The main question there is: Which finite groups are $\frac{3}{2}$-generated? $\endgroup$ – Dietrich Burde May 3 '17 at 14:44

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, $1995$.

  • $\begingroup$ Thank you. That $S_n$ is $\tfrac{3}{2}$-generated for $n>4$ has been apparently proven by G. Ya. Binder, The bases of the symmetric group. But that article is in Russian, so that I cannot read it. I have only found that reference in Brenner, Wiegold, Two-generator groups, I. $\endgroup$ – HeinrichD May 3 '17 at 14:54
  • $\begingroup$ I believe I.M. Isaacs wrote an article in the AMM a few years ago with a proof of this fact (for $S_n$). I'll try and dig up the reference. $\endgroup$ – Steve D May 3 '17 at 18:39
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    $\begingroup$ Yes, it is: Isaacs, I.M.; Zieschang, Thilo. Generating Symmetric Groups. AMM 102 (1995) 734-739 $\endgroup$ – Steve D May 3 '17 at 18:47
  • $\begingroup$ @SteveD Many thanks! I have added the links. $\endgroup$ – Dietrich Burde May 3 '17 at 18:53

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