torsion group

The Brauer group of any field is a torsion group.

The following curves have a torsion group isomorphic to :

So the torsion group of an Edwards curve over is isomorphic to either or .

The latter are just the abelian torsion groups.

Obviously, every finite abelian group is a torsion group.

In a sense, this means that studying p-torsion groups in isolation tells us everything about torsion groups in general.

A group G is a torsion group if every element in G is of finite order.

In group theory, a periodic group or a torsion group is a group in which each element has finite order.

For Dedeking domains with all quotient by primes ideals finite, SK is a torsion group.

Moreover, elliptic curves whose Mordell-Weil groups over Q have the same torsion groups belong to a parametrized family.

The category of abelian torsion groups is a coreflective subcategory of the category of abelian groups.

In mathematics, a torsion sheaf is a sheaf on a site for which, for every quasi-compact U, the space of sections is a torsion group.

Primary cyclic groups are characterised among finitely generated abelian groups as the torsion groups that cannot be expressed as a direct sum of two non-trivial groups.

Nor need there be an upper bound on the orders of elements in a torsion group if it isn't finitely generated, as the example of the factor group Q/Z shows.

In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group.

Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsion-free groups, exemplified by the groups Q/Z (periodic) and Q (torsion-free).

For C, and hence by the Lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to (Q/Z).

For example, the "cyclic group" (meaning that it is generated by one element - not to be confused with a torsion group) generated by a rotation by an irrational number of turns about an axis.

It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion.

There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup.

Not every torsion group is finite however: consider the direct sum of a countable number of copies of the cyclic group C; this is a torsion group since every element has order 2.

The Baer Fomin Theorem states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group, that is, a group of bounded exponent.

In mathematics, the torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field.

The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of p-torsion groups.