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I have two groups $(Z_2)^n$ equiped with a pointwise binary addition operator and $S_n$ acting iteratively on the space $X=\{0,1\}^n$, in the following way: Fix $x\in X$. At each iteration we apply a transposition from $S_n$ to $x$ and then an element of $Z_2^n$ acts on the result, and so on. For example, fix $(1,0,0)\in X$, then for $(13)\in S_3$ and $(1,1,0)\in Z_2^3$, we have

$$(1,1,0) \circ ((13) \circ (1,0,0)) = (1, 1,0) \circ (0, 0,1) = (1,1,1)$$

Now choose other elements from $S_3$ and $Z_2^3$ And apply them to $(1,1,1)$. Continue this process.

How can one study this iterative process? Is there a way of mixing these two groups and define a group action?

RezaR
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  • So, like a Markov chain? – anon Apr 19 '18 at 00:40
  • Like an iteratative function system. Think about the action of each group as a function. – RezaR Apr 19 '18 at 00:57
  • So is the choice of transposition and translation vector fixed ahead of time? – anon Apr 19 '18 at 00:59
  • No. Assume that is done randomly, so it could be thought as a stochastic process that is obtained by sequential actions of two different groups to a fix point x. – RezaR Apr 19 '18 at 01:04

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