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Added a note that the list is complete up to |G| = 10^18.
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Stefan Kohl ♦
Stefan Kohl ♦
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For abelian simple groups your question is merely a disguised form of "enumerate the Mersenne primes". The smallest examples for nonabelian simple groups are as follows:

  • $|{\rm PSL}(2,7)| + 1 = 13^2$,

  • $|{\rm A}_6| + 1 = 19^2$,

  • $|{\rm M}_{11}| + 1 = 89^2$,

  • $|{\rm PSU}(4,2)| + 1 = 161^2$,

  • $|{\rm J}_1| + 1 = 419^2$

These are all examples with order $\leq 10^{18}$.

For abelian simple groups your question is merely a disguised form of "enumerate the Mersenne primes". The smallest examples for nonabelian simple groups are as follows:

  • $|{\rm PSL}(2,7)| + 1 = 13^2$,

  • $|{\rm A}_6| + 1 = 19^2$,

  • $|{\rm M}_{11}| + 1 = 89^2$,

  • $|{\rm PSU}(4,2)| + 1 = 161^2$,

  • $|{\rm J}_1| + 1 = 419^2$

For abelian simple groups your question is merely a disguised form of "enumerate the Mersenne primes". The smallest examples for nonabelian simple groups are as follows:

  • $|{\rm PSL}(2,7)| + 1 = 13^2$,

  • $|{\rm A}_6| + 1 = 19^2$,

  • $|{\rm M}_{11}| + 1 = 89^2$,

  • $|{\rm PSU}(4,2)| + 1 = 161^2$,

  • $|{\rm J}_1| + 1 = 419^2$

These are all examples with order $\leq 10^{18}$.

Source Link
Stefan Kohl ♦
Stefan Kohl ♦
  • 19.6k
  • 21
  • 75
  • 137

For abelian simple groups your question is merely a disguised form of "enumerate the Mersenne primes". The smallest examples for nonabelian simple groups are as follows:

  • $|{\rm PSL}(2,7)| + 1 = 13^2$,

  • $|{\rm A}_6| + 1 = 19^2$,

  • $|{\rm M}_{11}| + 1 = 89^2$,

  • $|{\rm PSU}(4,2)| + 1 = 161^2$,

  • $|{\rm J}_1| + 1 = 419^2$