Problem For Week Nine

Problem Nine (Jan 19th): Let $V=R^n$ with standard orthorgonal basis $\epsilon_1,\epsilon_2,\cdots,\epsilon_n$ , and define $\Phi$ to be the set of all $2n(n-1)$ vectors $\pm\epsilon_i\pm\epsilon_j, (1\leq i<j\leq n)$ . Let W be the group generated by the $2n(n-1)$ reflections in $R^4$ sending $\alpha = \pm\epsilon_i\pm\epsilon_j$ to its negative while fixing pointwise the hyperplane that is orthorgonal with $\alpha$ . Prove that W is the semidirect product of the symmetric group $S_n$ and $\Bbb Z_2^{n-1}$ .

Solutions to last problem: $p$ be a prime, and let $\Bbb Z_p=\Bbb Z / p\Bbb Z$ denote the congruence classes in $\Bbb Z$ modulo $p$ . $\Bbb Z_p^*=\Bbb Z_p-\{0\}$ also admits multiplication, making $\Bbb Z_p$ a finite field.

Now we fix a $n$ and define $G_n=\{x^n:x\in \Bbb Z_p^*\}$ . Then $G_n$ is a subgroup of $\Bbb Z_p^*$ , and we can split $\Bbb Z_p^*$ as coset space $\Bbb Z_p^*=a_1G_n\cup a_2G_n\cup\cdots\cup a_rG_n$ .

Since $z\to z^n$ is a surjective homomorphism from $\Bbb Z_p^*$ to $G_n$ , we have from the the fundamental theorem on homomorphisms that $r$ is the number of solutions of the function $x^n=1$ in $\Bbb Z_p^*$ .

When $x \in a_i G_n$ , color x with $i$ . This gives an $r$ coloring of $\Bbb Z_p^* = \{1, 2,\cdots, p - 1\}$ . If $p-1 \geq S(n) \geq S(r)$ , then by theorem 2 there is a monochromatic Schur triple, that is, there are integers $x, y, z$ , none of which is divisible by p, such that $a_ix^n + a_iy^n \equiv a_iz^n \mod p$ , since $a_i$ is not divisible by $p$ it follows that $x_n + y_n = z_n \mod p$ , completing the proof.

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One Comment on “Problem For Week Nine”

Anonymous Says:

2011/03/09 at 9:41 pm
\text{Learn} \text{to} \text{use} \epsilon -\delta \text{language}, \text{and} \text{you} \text{will} \text{achieve} \text{more}.\text{See} \text{this} \text{equation}:\int x^x \, dx=\sum _{n=1}^{\infty } x^?

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