Skip to main content
MathOverflow

Return to Answer

rearranged text and clarified
Source Link
Geoff Robinson
Geoff Robinson
  • 44.4k
  • 5
  • 123
  • 169

The original question has been answered. As background information, I point out that S. Amitsur classified finite groups which can occur as subgroups of multiplicative groups of divison algebras. These include finite groups which have a faithful "one-dimensional" representation over the quaternions. As noted in Ben's answer, any such finite group (in the general division ring case) has cyclic Sylow $p$-subgroups for each odd prime $p$ and cyclic or generalized quaternion Sylow $2$-subgroups. It also satisfies other group-theoretic properties, such as elements of order $p$ and $q$ commuting when $p$ and $q$ are distinct primes. 

In fact, finite groups which occur as subgroups of multiplicative groups of division algebras are precisely the Frobenius complements. NoteWe note that any generalized quaternion group already occurs (- this is not in conflict with Ben's answer: if that group has order greater than $8,$ then it has a non-Abelian dihedral group as a homomorphic image, so not all its irreducible representations over the quaternions are $1$-dimensional). The only non-solvable group which occurs is ${\rm SL}(2,5),$ which does already occur inside the multiplicative group of the real quaternions ( as does each generalized quaternion $2$-group)- again, the fact that ${\rm SL}(2,5)$ has $A_{5}$ as a homomorphic image means that not all its irreducible representations over the quaternions are $1$-dimensional.

The original question has been answered. As background information, I point out that S. Amitsur classified finite groups which can occur as subgroups of multiplicative groups of divison algebras. These include finite groups which have a faithful "one-dimensional" representation over the quaternions. As noted in Ben's answer, any such finite group (in the general division ring case) has cyclic Sylow $p$-subgroups for each odd prime $p$ and cyclic or generalized quaternion Sylow $2$-subgroups. It also satisfies other group-theoretic properties, such as elements of order $p$ and $q$ commuting when $p$ and $q$ are distinct primes. In fact, finite groups which occur as subgroups of multiplicative groups of division algebras are precisely the Frobenius complements. Note that any generalized quaternion group occurs ( this is not in conflict with Ben's answer: if that group has order greater than $8,$ then it has a non-Abelian dihedral group as a homomorphic image, so not all its irreducible representations over the quaternions are $1$-dimensional). The only non-solvable group which occurs is ${\rm SL}(2,5),$ which does already occur inside the multiplicative group of the real quaternions ( as does each generalized quaternion $2$-group)- again, the fact that ${\rm SL}(2,5)$ has $A_{5}$ as a homomorphic image means that not all its irreducible representations over the quaternions are $1$-dimensional.

The original question has been answered. As background information, I point out that S. Amitsur classified finite groups which can occur as subgroups of multiplicative groups of divison algebras. These include finite groups which have a faithful "one-dimensional" representation over the quaternions. As noted in Ben's answer, any such finite group (in the general division ring case) has cyclic Sylow $p$-subgroups for each odd prime $p$ and cyclic or generalized quaternion Sylow $2$-subgroups. It also satisfies other group-theoretic properties, such as elements of order $p$ and $q$ commuting when $p$ and $q$ are distinct primes. 

In fact, finite groups which occur as subgroups of multiplicative groups of division algebras are Frobenius complements. We note that any generalized quaternion group already occurs - this is not in conflict with Ben's answer: if that group has order greater than $8,$ then it has a non-Abelian dihedral group as a homomorphic image, so not all its irreducible representations over the quaternions are $1$-dimensional. The only non-solvable group which occurs is ${\rm SL}(2,5),$ which does already occur inside the multiplicative group of the real quaternions ( as does each generalized quaternion $2$-group)- again, the fact that ${\rm SL}(2,5)$ has $A_{5}$ as a homomorphic image means that not all its irreducible representations over the quaternions are $1$-dimensional.

Source Link
Geoff Robinson
Geoff Robinson
  • 44.4k
  • 5
  • 123
  • 169

The original question has been answered. As background information, I point out that S. Amitsur classified finite groups which can occur as subgroups of multiplicative groups of divison algebras. These include finite groups which have a faithful "one-dimensional" representation over the quaternions. As noted in Ben's answer, any such finite group (in the general division ring case) has cyclic Sylow $p$-subgroups for each odd prime $p$ and cyclic or generalized quaternion Sylow $2$-subgroups. It also satisfies other group-theoretic properties, such as elements of order $p$ and $q$ commuting when $p$ and $q$ are distinct primes. In fact, finite groups which occur as subgroups of multiplicative groups of division algebras are precisely the Frobenius complements. Note that any generalized quaternion group occurs ( this is not in conflict with Ben's answer: if that group has order greater than $8,$ then it has a non-Abelian dihedral group as a homomorphic image, so not all its irreducible representations over the quaternions are $1$-dimensional). The only non-solvable group which occurs is ${\rm SL}(2,5),$ which does already occur inside the multiplicative group of the real quaternions ( as does each generalized quaternion $2$-group)- again, the fact that ${\rm SL}(2,5)$ has $A_{5}$ as a homomorphic image means that not all its irreducible representations over the quaternions are $1$-dimensional.