I'm having a bit of problem trying to find isomorphisms between the following groups:
$D_6$, $A_4$, $S_3 \times \Bbb{Z}_2$, and $G$.
G is a group generated by $a, b, c$ which follow these rules: $a^2=b^2=c^3=id$ (id = identity), $ca=bc$, $cb=abc$, $ab=ba$.
I can find isomorphisms between...
Hello,
I've been using Caratheodory's Lemma to prove the Inverse Function Theorem and Fermat's Theorem. I have managed to prove both of them, I would just like someone to look over my proof and tell me if I'm missing anything (i.e. should I clarify any parts of my proof). So here goes:
Inverse...
In class we had to show that ${A}_{5}$ is cyclic. So what we did was,
${A}_{5}$ is cyclic iff there is an $\alpha\in{A}_{5}$ with $<\alpha> = {A}_{5}$. So, the $ord(\alpha) = |<\alpha>| = |{A}_{5}| = \frac{5!}{2} = 60$. So, $60 = {2}^{2}*3*5$.
After this, we said that we could do a 4-cycle...
Oh, I apologize. So then that would be why they did $(6*3)+(3*6)$. I see it now. Well could we have done the numerator in another way?
EDIT: OH They have another way to do the numerator which is $(3*3!)+(3*3!)$ which comes out to the same result.
I believe the xyz part is where I'm getting messed up at. I can see 3 but do I multiply it by 2 because we have two different outcomes? That is we have the first situation being [2|4] and the other in [3|6]? If that's the case then we'd have 6.
I know this should be easy to understand but I just need a little clarification on the last part of my answer for this problem:
If three distinct dice are rolled, what is the probability that the highest value is twice the smallest value
I started this problem with the understanding that there...
I'm given the following Piecewise function when $f:[0,1]\to[0,1]$:
$f(x) = x$ when $x\in\Bbb{Q}$
$f(x) = 1-x$ when $x\notin\Bbb{Q}$
I need to prove that $f$ is continuous only at the point $x=\frac{1}{2}$.
For this problem, I know I need to use the fact that a function $f$ is continuous at a...
So I was pulling my hair out at Barnes and Noble doing this problem because I was completely unsure if I could just use Cauchy Sequences to prove this problem since my textbook is mean and likes to not name certain things. I decided to go back and look at some of the previous problems in the...
I was just looking through my textbook to brush up on Cauchy sequences since I was planning on using it for this problem but my book utterly leaves it out. But I did come across this Theorem:
Suppose that f is a function and f:S\to\Bbb{R}. If f is uniformly continuous on S then, given any two...
Ah yes, I apologize. It is $\overline{S}$. Thank you very much for the hint! I don't recall covering Cauchy Sequences in my intro class so I will have to read up on it but I do see where I can go with it now.
Hello,
I've been attempting to do these problems from my textbook:
1. Suppose that f is a continuous function on a bounded set S. Prove that the
following two conditions are equivalent:
(a) The function f is uniformly continuous on S.
(b) It is possible to extend f to a continuous function on...