# Find elements of left coset ZZ_6/{0,3}

If I have a group G = (ZZ_6,+), i.e. the integers modulo 6 under addition, and a subgroup of G, H = ({0,3}, +), how can I find G/H with sage?

Find elements of left coset ZZ_6/{0,3}

add a comment

1

Note that the sage instances corresponding to the *ring* $(\Bbb Z/6,+,\cdot)$, and to the cyclic *abelian group* $(C_6,+)$ differ, as it also happens in the mathematical world. The quotient in the category of rings is built w.r.t. an ideal, in our case the ideal is generated by $3\in \Bbb Z/6$. In the category of abelian groups we mod out a subgroup, in our case the subgroup is generated by $3$. Let us construct the quotient in both cases.

In the category of rings, we introduce the ring $R$ by either of...

```
sage: Integers(6)
Ring of integers modulo 6
sage: Zmod(6)
Ring of integers modulo 6
sage: IntegerModRing(6)
Ring of integers modulo 6
```

then consider its ideal $J=(3)$, and the quotient:

```
sage: R = Zmod(6)
sage: R
Ring of integers modulo 6
sage: J = R.ideal(3)
sage: J
Principal ideal (3) of Ring of integers modulo 6
sage: list(R)
[0, 1, 2, 3, 4, 5]
sage: Q = R.quotient(J)
sage: Q
Ring of integers modulo 3
sage: for k in R:
....: print(f"{k} modulo J is {Q(k)}")
....:
0 modulo J is 0
1 modulo J is 1
2 modulo J is 2
3 modulo J is 0
4 modulo J is 1
5 modulo J is 2
sage:
```

In the category of abelian groups (and/or in the category of groups) one can construct:

```
sage: C6.<a> = AbelianGroup(1, [6])
sage: C6
Multiplicative Abelian group isomorphic to C6
sage: # or
sage: C6 = CyclicPermutationGroup(6)
sage: a = C6.gens()[0]
sage: a.order()
6
sage: a
(1,2,3,4,5,6)
```

And the quotient is...

```
sage: H = C6.subgroup([a^3])
sage: Q = C6.quotient(H)
sage: Q
Permutation Group with generators [(1,2,3)]
```

Unfortunately, the version using `C6.<a> = AbelianGroup(1, [6])`

was leading to a sage crash on my machine while trying to build the quotient with respect to the subgroup `H`

constructed mot-a-mot as above. So just use the construction involving a permutation group.

Asked: **
2020-06-19 00:28:22 -0600
**

Seen: **51 times**

Last updated: **Jun 29 '20**

How to iterate over groups in increasing size

Group given by congruence relation

Cosets Generated by Product of Generators

Faster function for working with cosets

Is there any way to find decomposition group and ramification groups

How do I identify the set of words of given length in matrix generators

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.