Table of Contents

## How to create a multiplicative abelian group in Sage?

To construct this Abelian group in Sage, you can either specify all entries of →k or only the non-zero entries together with the total number of generators: It is also legal to specify 1 as the order. The corresponding generator will be the neutral element, but it will still take up an index in the labelling of the generators:

## How to use Gens _ orders in abelian groups?

You should now use gens_orders () instead: Background on invariant factors and the Smith normal form (according to section 4.1 of [C1]): An abelian group is a group A for which there exists an exact sequence Zk → Zℓ → A → 1 , for some positive integers k, ℓ with k ≤ ℓ. For example, a finite abelian group has a decomposition

## Which is the underlying representation of an abelian group?

An Abelian group uses a multiplicative representation of elements, but the underlying representation is lists of integer exponents: [C1] H. Cohen Advanced topics in computational number theory, Springer, 2000. [C2] —-, A course in computational algebraic number theory, Springer, 1996.

## Which is the matrix of a multiplicative abelian group?

The matrix of relations M: Zk → Zℓ is the matrix whose rows generate the kernel of ϕ as a Z -module. In other words, M = (Mij) is a ℓ × ℓ diagonal matrix with Mii = pcii. Consider now the subgroup B ⊂ A generated by b1 = af1, 11… afℓ, 1ℓ, …, bm = af1, m1… afℓ, mℓ .

## Which is not unique in a multiplicative abelian group?

Note that this presentation is not unique, for example Z6 ≅ Z2 × Z3. The orders of the generators →k = (0, …, 0, k1, …, kt) has previously been called invariants in Sage, even though they are not necessarily the (unique) invariant factors of the group.

## Which is an abelian group under the addition operation?

Thus the integers, , form an abelian group under addition, as do the integers modulo , /. Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group.

## Which is an isomorphic group to an abelian group?

Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is abelian. In fact, for every prime number p there are (up to isomorphism) exactly two groups of order p 2, namely Z p 2 and Z p×Z p.