I know that $S_n$ is generated by a number of things, like all transpositions, all transpositions of the form $(1\,a)$, the transpositions $(1\,2),(2\,3),\dots,(n-1\, n)$, and  just the two elements $(123\dots n),(1\,2)$.      Suppose $n$ is prime. If you just have $(123\dots n)$ and some arbitrary transposition $(a\, b)$, how does this also generate $S_n$?    Can you somehow get to $(1\,2)$ or reduce it to some other previous case?
I have a homework question that asks     Find necessary and sufficient conditions on $1 \leq i &lt; j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.   Here is what I have done so far.  Call $c = (1 \, 2 \, \dotsc \, n)$.  I made the observation that $$ \overbrace{c c \dotsb c}^{n - j + 1} (i \, j) \overbrace{c^{-1} c^{-1} \dotsb c^{-1}}^{n - j + 1} = (1 + i - j + n \, 1). $$ Thus, $\langle (i \, j), c \rangle = \langle (1 \, i - j + n + 1), c \rangle$, so it suffices to look at $\langle (1 \, k), c \rangle$ for positive integers $2 \leq k \leq n$.  I've checked that, for the value $k=2$, $\langle (1 \, 2), c \rangle = S_n$.  This immediately implies $\langle (1 \, n), c \rangle = S_n$.  I suspect that no other values of $k$ will work, but am not sure how to prove it.
What are integers $x$ and $n$ such as $ x^4+x^3+x^2+x+1 = n^2$? What are the integer numbers $x$ such as $x^4+x^3+x^2+x+1$ is the square of an integer?  $0$ and $-1$ are obvious answers, but how many are there? I'm trying to get a Schengen Visa from my first port of entry, France. I am trying to book an appointment with the French Consulate in SF, and the URL I am using is the following:   I've been checking this page periodically for the past few weeks, and I always see a dialog that that says "We are currently at full capacity. Please try again later." Their calendar provides no direct option for going to the next month. (Using the in-browser developer tools, I noticed that the button for next month is hidden. Un-hiding it and clicking on the Next Month button displays no available appointments either. I checked all the way until September 2016).  My guess is that there are absolutely no appointments for the next 6 months. Or, more likely, their page isn't functioning properly. 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So what if we do the same thing, but a different matrix? $$ z = a + bh$$ can describe a matrix $$ \begin{bmatrix} a &amp; b \\ b &amp; a\end{bmatrix} $$ and we can discover it the same way $$ \begin{pmatrix} a \\ b \end{pmatrix} \cdot \begin{bmatrix} c &amp; d \\ d &amp; c\end{bmatrix} = \begin{pmatrix} ac + bd \\ ad + bc \end{pmatrix} $$ $$ (a + bh) \cdot (c + dh) = ac + adh + bch + bdh^2 $$ From here we infer that $ h^2 = 1 $.  Also $$ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots  $$ $$ \begin{align} e^{xh} &amp; = 1 + \frac{xh}{1!} + \frac{(xh)^2}{2!} + \frac{(xh)^3}{3!} + \frac{(xh)^4}{4!} + \frac{(xh)^5}{5!} + \cdots \\ &amp; = 1 + \frac{xh}{1!} + \frac{x^2}{2!} + \frac{x^3h}{3!} + \frac{x^4}{4!} + \frac{x^5h}{5!} + \cdots \\ &amp; = \cosh{x} + h \cdot \sinh{x} \end{align} $$  How many more numbers like this are there? 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If I want to prove that $\mathbb{R} \setminus \mathbb{Q}$ is disconnected, does it suffice to say that there are two open disjoint sets that cover $\mathbb{R}\setminus\mathbb{Q}$, namely:  $$(- \infty, 0), (0, \infty)\text{ ?}$$  Along the same lines, I want to prove or disprove that $S = \mathbb{R}^2 \setminus \mathbb{Q}^2$ (points $(x, y) \in S$ that have at least one irrationais connected. I feel that it is also disconnected; does it suffice to say that there are two open disjoint sets that cover $S = \mathbb{R}^2 \setminus \mathbb{Q}^2$, namely $((- \infty, - \infty), (0, 0))$ and $((0, 0), (\infty, \infty))$? (Very iffy on my assertion and my notation, sorry.)  Thanks! I want to run a daily incremental backup and append the day of the week to the file name so I end up with:  backup_mon.bak backup_tue.bak  etc.  Not really bothered about what exactly is appended - 0 - 6 is fine.  How do I go about this? What's the term for using the wrong word because it sounds the same? 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What does "proverbial" mean?