Return to Answer

One often reads that divisible groups are important since they help us understanding the structure of abelian groups, for they are all and the only injectives in the usual category of abelian groups (which is, of course, undeniable). Yet, I find that the non-abelian case is, if possible, even more interesting. A 'natural' example of a non-commutative divisible group is the group of units of Hamilton's quaternions; afak, the result is due to I. Niven [1]. On another hand, it was recently proved on this forum that the general linear group of degree $n$ over an algebraically closed field $\mathbb K$ is divisible iff $\mathbb K$ has zero characteristic (see here), and I've just posed a similar question for ${\rm SL}_n(\mathbb K)$ (see here)

[1] I. Niven, The Roots of a Quaternion, The Amer. Math. Monthly, Vol. 49, No. 6 (Jun. - Jul., 1942), pp. 386-388.

One often reads that divisible groups are important since they help us understanding the structure of abelian groups, for they are all and the only injectives in the usual category of abelian groups (which is, of course, undeniable). Yet, I find that the non-abelian case is, if possible, even more interesting. A 'natural' example of a non-commutative divisible group is the group of units of Hamilton's quaternions; afak, the result is due to I. Niven [1]. On another hand, it was recently proved on this forum that the general linear group of degree $n$ over an algebraically closed field $\mathbb K$ is divisible iff $\mathbb K$ has zero characteristic (see here), and I've just posed a similar question for ${\rm SL}_n(\mathbb K)$ (see here)

[1] I. Niven, The Roots of a Quaternion, The Amer. Math. Monthly, Vol. 49, No. 6 (Jun. - Jul., 1942), pp. 386-388.

One often reads that divisible groups are important since they help us understanding the structure of abelian groups, for they are all and the only injectives in the usual category of abelian groups (which is, of course, undeniable). Yet, I find that the non-abelian case is, if possible, even more interesting. A 'natural' example of a non-commutative divisible group is the group of units of Hamilton's quaternions; afak, the result is due to I. Niven [1]. On another hand, it was recently proved on this forum that the general linear group of degree $n$ over an algebraically closed field $\mathbb K$ is divisible iff $\mathbb K$ has zero characteristic (see here), and I've just posed a similar question for ${\rm SL}_n(\mathbb K)$: In particular, $\mathbb K$ having zero characteristic is still necessary, and under this assumption it is not difficult to prove that ${\rm SL}_2(\mathbb K)$ is divisible, but I don't know about the general case (see here).

[1] I. Niven, The Roots of a Quaternion, The Amer. Math. Monthly, Vol. 49, No. 6 (Jun. - Jul., 1942), pp. 386-388.

One often reads that divisible groups are important since they help us understanding the structure of abelian groups, for they are all and the only injectives in the usual category of abelian groups (which is, of course, undeniable). Yet, I find that the non-abelian case is, if possible, even more interesting. A 'natural' example of a non-commutative divisible group is the group of units of Hamilton's quaternions; afak, the result is due to I. Niven [1]. On another hand, it was recently proved on this forum that the general linear group of degree $n$ over an algebraically closed field $\mathbb K$ is divisible iff $\mathbb K$ has zero characteristic (see here), and I've just posed a similar question for ${\rm SL}_n(\mathbb K)$ (see here)

[1] I. Niven, The Roots of a Quaternion, The Amer. Math. Monthly, Vol. 49, No. 6 (Jun. - Jul., 1942), pp. 386-388.

One often reads that divisible groups are important since they help us understanding the structure of abelian groups, for they are all and the only the injectives in the usual category of abelian groups (which is, of course, undeniable). Yet, I find that the non-abelian case is, if possible, even more interesting. A 'natural' example of a non-commutative divisible group is the group of units of Hamilton's quaternions (afak, the result is due to I. Niven). On another hand, it was recently proved on this forum that the general linear group of degree $n$ over an algebraically closed field $\mathbb K$ is divisible iff $\mathbb K$ has zero characteristic (see here), and I've just posed a similar question for ${\rm SL}_n(\mathbb K)$: In particular, $\mathbb K$ having zero characteristic is still necessary, and under this assumption it is not difficult to prove that ${\rm SL}_2(\mathbb K)$ is divisible, but I don't know about the general case (see here).

One often reads that divisible groups are important since they help us understanding the structure of abelian groups, for they are all and the only injectives in the usual category of abelian groups (which is, of course, undeniable). Yet, I find that the non-abelian case is, if possible, even more interesting. A 'natural' example of a non-commutative divisible group is the group of units of Hamilton's quaternions; afak, the result is due to I. Niven [1]. On another hand, it was recently proved on this forum that the general linear group of degree $n$ over an algebraically closed field $\mathbb K$ is divisible iff $\mathbb K$ has zero characteristic (see here), and I've just posed a similar question for ${\rm SL}_n(\mathbb K)$: In particular, $\mathbb K$ having zero characteristic is still necessary, and under this assumption it is not difficult to prove that ${\rm SL}_2(\mathbb K)$ is divisible, but I don't know about the general case (see here).

[1] I. Niven, The Roots of a Quaternion, The Amer. Math. Monthly, Vol. 49, No. 6 (Jun. - Jul., 1942), pp. 386-388.