Zulip Chat Archive

Stream: maths

Topic: normal subgroup : group :: ?? : semimodule


view this post on Zulip Kenny Lau (Jul 06 2020 at 08:42):

Normal subgroups are the kernels of group homomorphisms. More precisely, a subset of a group is a normal subgroup iff it is the kernel of some group homomorphism.

Then how do we characterise kernels of semimodule homomorphisms? Obviously the kernel of a semimodule homomorphism is a submodule; but given any submodule can we produce a homomorphism with the specified kernel?

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:43):

(btw the double colon means analogy)

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:44):

For example, consider (a,b)ab:N2Z(a,b) \mapsto a-b: \mathbb{N}^2 \to \mathbb{Z}, a homomorphism between N\mathbb{N}-semimodules. Its kernel is {(n,n)nN}\{(n,n) \mid n \in \mathbb{N}\}.

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:45):

Given this kernel, how do we recover Z\mathbb{Z}?

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:47):

Maybe given kernel SS we should define ab:=sS,b=a+sa \sim b := \exists s \in S, b=a+s

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:47):

good luck proving that this relation is symmetric

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:48):

what if we define ab:=sS,b=a+sa=b+sa \sim b := \exists s \in S, b=a+s \lor a=b+s (good luck proving transitivity)

view this post on Zulip Chris Hughes (Jul 06 2020 at 08:48):

The kernel is a relation. Two homomorphisms with the same kernel do not have isomorphic images necessarily. e.g. The multiplicative map from N->N sending even numbers to zero and odd numbers to themselves has kernel {1} but is not the identity.

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:49):

your map isn't linear

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:50):

aha, we're in multiplicative land

view this post on Zulip Kenny Lau (Jul 06 2020 at 08:51):

ok, then what are n and s conditions on the relation to ensure that it is the kernel of some map?

view this post on Zulip Chris Hughes (Jul 06 2020 at 08:52):

It has to be a subsemimodule of the product.

view this post on Zulip Chris Hughes (Jul 06 2020 at 08:53):

i.e. it must "preserve" all the operations so I can define them on the quotient.

view this post on Zulip Chris Hughes (Jul 06 2020 at 08:54):

@Amelia Livingston did all this for monoids. Not for semimodules yet.

view this post on Zulip Chris Hughes (Jul 06 2020 at 09:01):

multiset.card is a better example of a map with trivial kernel that's not injective.

view this post on Zulip Kevin Buzzard (Jul 06 2020 at 09:35):

The additive map from nat to fin (n+1) sending all big numbers to n is another example (where the kernel doesn't determine the map). Amelia taught me how to think about this, although doubtless it was well known beforehand. A surjection of sets/types XYX\to Y, up to isomorphism, is just an equivalence relation on XX. And now when you start adding structure, a surjection of groups/rings/whatever is just an equivalence relation which plays well wrt the structure. Analysing the lattice of normal subgroups of a group is just the same as analysing the lattice of isomorphism classes of quotients of the group. The quotient story is the one which generalises

view this post on Zulip Chris Hughes (Jul 06 2020 at 09:57):

This came up in my Group Theory course. There was this yucky definition of "block"s of a group action. Let GG be a group acting on a set Ω\Omega. A non-empty subset BB of Ω\Omega is said to be a block if for each gGg \in G, either gB=BgB=B or gBB=gB \cap B = \emptyset. Everything became much easier when I realised a block is just an equivalence class for an equivalence relation with the property ab    g,gagba \sim b \implies \forall g, ga \sim gb, or equivalently the preimage of a point under a morphism of GG-sets.

view this post on Zulip Yury G. Kudryashov (Jul 06 2020 at 18:35):

We have con and add_con. I hope eventually they will be used to define quotient groups etc


Last updated: May 18 2021 at 07:19 UTC