Intereting Posts

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let $\mathbf{A}$ be the set of all $n\times n$ matrices on $\mathbb{N}_{n^2}=\{1,2,…,n^2\}$ with distinct entries.

let $T$ be the set of all permutations of $\mathbf{A}$ which swap the entry 1 with one of its adjacent entries. adjacent means one is above/below/right/left of the other (and both are neighbors).

Let $S$ be the permutation subgroup generated by $T$.

- How to enumerate subgroups of each order of $S_4$ by hand
- Adapting a proof on elements of order 2: from finite groups to infinite groups
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- Group of order $|G|=pqr$, $p,q,r$ primes has a normal subgroup of order
- Showing $G/Z(R(G))$ isomorphic to $Aut(R(G))$
- Existence of normal subgroups for a group of order $36$

Show that $S$ acts ** intransitively** on $\mathbf{A}$.

I’m a.s. it’s correct for even $n$.

- “Natural” example of cosets
- How to enumerate subgroups of each order of $S_4$ by hand
- Find the center of the symmetry group $S_n$.
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- Proof involving Cyclic group, generator and GCD
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- Is there a convenient way to show that the symmetric group $S_4$ has a subgroup of order $d$ for each $d|24$?

This appears to be a generalization of the famous 15-puzzle. The proof that it is not transitive is the same. To each matrix, associate the number given by

\begin{equation}

\text{(block distance of 1 from upper left corner)} +

\text{(parity of underlying permutation of $\mathbb{N}_{n^2}$)}.

\end{equation}

The parity of this number is an invariant under the allowed moves, that is, it does not change when you make a swap. So if you start with the numbers neatly ordered row by row, you will never get the matrix with 2 and 3 exchanged.

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