What is known about this type of group?

[Someone2841]

Hello. I have been looking into group theory for its applications to subject I am studying. I am not a mathematician by profession or training, but I find it has great use to any analytical pursuit. With that said, I have outlined below type of group that I would like to know more about. For example: what other groups are like it?, is there anything strange about the way I've defined this group?, what would help in my understanding of this group and groups like it?, etc. Thanks in advance!

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Let $C$ be the cyclic ring of integers mod 12. Please note that arithmetic with elements of $C$ and its subsets will be assumed to be mod 12 unless otherwise indicated. Define the set $M_{k \in C}$ as

$\{x : n \in \mathbb{Z}, k-1 \leq n \leq k+5, x \equiv 7n\}$​

For example, $M_0 = \{0,2,4,5,7,9,11\}$. These subsets $M_k$ clearly do not form subgroups of $C$; however, a group structure can be imposed on them. Define the action $S_k$ on $M_k$ as

$S_k:M_k×M_k,m≡7(x+k−1)↦7((x+_7 1)+k−1)$**​

Where $+_7$ is addition mod 7. For example, $S_0$ would map 4 onto 11 since $x=3$ in the equation $4 \equiv 7(x + k - 1)$ and $7((x +_7 1) + k - 1) \equiv 11$. Two iterations of the action (i.e., $S^2_k$) would then map each element to the next element (e.g., $9 \mapsto 11, 11 \mapsto 0, 0 \mapsto 2,$ etc. in $M_0$).

The set $\{S^n_k\}_{n \in Z, 0 \leq n \leq 6}$ forms a cyclic action group (with $S_k$ as the group generator) on $M_k$. The identity action is $S^0_k \doteq S^7_k$ and the inverse action is $S^{-n \mod{7}}_k$.

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**Edited from x-k+1 to x+k-1

 

[micromass]

May I ask what $11$ gets mapped to under $S_0$?

 

[Someone2841]

May I ask what $11$ gets mapped to under $S_0$?

I made a mistake. $S_k$ should be defined as

$S_k:M_k×M_k,m≡7(x+k-1)↦7((x+_7 1)+k-1)$​

in order to deal with the very case of $S_0(11)$. In the equation $11 \equiv 7(x + k - 1)$, $x = 6$ and so $7((x+_7 1)+k-1) \equiv 5$. With this correction, $11$ gets mapped to $5$ under $S_0$. Sorry for the mistake!

 

[Stephen Tashi]

is there anything strange about the way I've defined this group?,

Define the action $S_k$ on $M_k$ as

The only context that I'm familiar with for "actions" on sets is that there is a definition for a group action on a set. (Google tells me that there is also definition of a ring action - and a neat definition of a module: http://mathoverflow.net/questions/100565/why-are-ring-actions-much-harder-to-find-than-group-actions. ) To define a group action, you'd normally start with the group and show how it acts on a set. I think what you should do is define the "functions" $S_k$ and prove the set of functions forms a group. The abstract group that it forms then has the "group action" on the set specified by the functions.

I don't know what part of your result depends only on having a ring-with-identity, a commutative ring with identity, or a field. You might think about whether a similar construction works when applied to mod 6 arithmetic.

An abstract and uspecific way to ask questions about your example is:

We have a finite group (or a ring) G. G acts on its own elements. As a group of functions, G may not contain enough functions to implement all possible permutations of those elements. Suppose we have a proper subset M of the elements of G and we define a set S of mappings of M onto itself. "in terms of" the operations of G. Further suppose that this set of functions S forms a group under the operation of composing functions. Must the group S be isomorphic to some subgroup (or subring) of G ? (i.e. can we implement the functions in the group S by functions that define the action of G on its elements?) Or is it possible to do something "fundamentally new" by creating an S in such a manner.

I'd guess such questions have been investigated, but I don't know good keywords for searching for results.

 

[blue_raver22]

Hello, thank you for your interest in group theory and its applications. From what I can gather, this type of group is a cyclic action group on the set $M_k$, where $M_k$ is a subset of the cyclic ring of integers mod 12. This type of group is defined by the action $S_k$, which maps elements of $M_k$ to other elements within the subset using addition mod 7. The set $\{S^n_k\}_{n \in Z, 0 \leq n \leq 6}$ forms a cyclic action group, with $S_k$ as the group generator. It seems that this type of group has a specific structure and properties, and I am not aware of any other groups that are exactly like it. However, it may be helpful to explore other cyclic action groups and their properties to gain a better understanding of this group and how it relates to others. It may also be useful to further investigate the inverse and identity actions within this group to see how they affect the elements of $M_k$. I hope this helps in your understanding of this type of group and its potential applications.