Let $G$ be a group and $g \in G$. Show that $o(g)=o(xgx^{-1})$, where $o(x)$ is the order.  I've managed to show that the order of $xgx^{-1}$ is at most $o(g)$ by the following way. Denote $o(g) = n$ and $o(xgx^{-1}) = m.$ Now $(xgx^{-1})^n=xgx^{-1} \cdots xgx^{-1} = xg^nx^{-1} = xx^{-1} = e.$ But I'm not sure how to show this the other way around? Any hints would be appreciated. This doesn't seem to yet imply that the orders would equal one another.
I am trying to prove that any conjugate of $a$ has the same order as $a$.  Let $G$ be a group and let $a \in G$.  An element $b \in G$ is called a conjugate of $a$ if $b=xax^{-1}$.  My professor gave us a hint.  Prove $a^k=e$ iff $b^k=e$ for all $k \in \mathbb{Z}$ implies $o(a) = o(b)$.  My first move is to $b^k = (xax^{-1})^k = (x^{-1})^ka^kx^k = (x^{-1})^kex^k = (x^{-1})^kx^k=x^{-k+k}=x^0$  I am not sure that I am doing this correctly?  I am trying to get $b^k=e$.  However when I raise $(xax^{-1})^k$, I am not sure if I what I have done is correct?  I am kind of confused here.
Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$ Similar questions have been asked, but none with the given information. My textbook doesn't give me the fact that  $∂A = \text{Cl}(A) − \text{Int}(A)$. If that were the case, I could just state that definition and note that it's the intersection of closed sets.   I have very little knowledge of set theory and proofs, so I'm not sure how else to prove this. As always, I appreciate any help.  The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa.   "Let $A$ be a subset of a topological space $X$. Prove that $∂A$ is closed given  $∂A = \text{Cl}(A) − \text{Int}(A)$" Show that $G$ is cyclic Here is a problem from Herstein.     Let G be a finite abelian group so that the equation $x^n=e$ has at most $n$ solutions in $G$ for every positive integer $n$. Show that $G$ is cyclic.   I will have to basically show that there exists an element $g$ in $G$ having order $o(G)$.   I showed that such an element exists if $o(G)=p_1p_2...p_k$ where $p_i$ are primes. But I could not proceed if $o(G)=p_1^{a_1}p_2^{a_2}$. I feel that if I can prove this I will be able to extend. Access POST values in Symfony2 request object I need to group 400 points in 10 groups with 40 points each. The goal is to get the 40 points geographicly together like this:        Any idea how to do this with QGIS? Punctuation in an indirect quotation Consider the brief passage:     "I love your work, but calling you 'the artist' {1} just doesn't seem to be   cutting it anymore. What shall I call you?"      "OK. Call me George." {2}   So my instincts tell me to put {1} in single quotes as opposed to no punctuation (or perhaps even italics) but to do the opposite for {2}. Is this right? Why?  Let me expand this one further iteration:     "Wow, that's a great name, especially with this crowd. Maybe for your next book    you should call yourself Ringo."{3} How to find the sum $$\sum^n_{k=1} (k^2+k+1)k!$$  What I tried as follows :   $$\sum^n_{k=1} (k^2+k+1)k!$$  =$$\sum^n_{k=1} (k^2)k!+ \sum^n_{k=1} (k)k! + \sum^n_{k=1} (1)k!$$  Now we can write $\sum^n_{k=1} (k^2) $ as $$\frac{n(n+1)(2n+1)}{6}$$ but what to do with $k!$ please guide thanks....... Are there network-wide reputation leagues? I would love to see which people have the most network-wide reputation, how can I see that? How to extract code of .apk file which is not working? Cross tabulation of two categorical variables: recommended techniques MT7610U - Unable to install wifi driver Has confinement been experimentally observed? So, confinement has obviously been shown by lattice gauge theory to be a predicted aspect of QCD. However, to what extent has it been observed in experimental physics? How do I install pdfedit? I'm running an Ubuntu 14.04 and want to install the Eclipse IDE for both Java and C/C++ programming. I installed the separate installation files from their official website. I followed this thread . (The top answer)   The steps helped me install the Java IDE but I don't know what changes I should make to the procedure to install the C/C++ IDE.   Another thread , recommended to install eclipse-cdt but the Eclipse version installed is outdated. (Doesn't allow me to install Pydev)   I have TWO questions:   Can the C/C++ libraries be downloaded from within eclipse, like Pydev? If not, how do I install both IDEs without interfering with each other?   P.S. I commented this query in the first thread but got no reply, that is why I'm asking a fresh question! "media change: please insert the disc labeled" when trying to install Ruby on Rails Reference style links validating as improperly formatted code Missing links to responses to comments on nomination posts In the responses tab in my profile, responses to comments on nomination posts include a working link in the page source, but because there's no title the links are empty and useless.    When listed in the inbox such posts do have a title, though in this case the link is to the post and not directly to the comment. (Actually, the links in responses end up being redirected and also go to the post and not to the comment, but they try to go to the comment.)  In Supercollider:    In Global Profile: How can I understand and prove the "sum and difference formulas" in trigonometry? Could someone advise me on how to determine all entire functions $f$ such that $|f'(z)| \leq |f(z)|, \forall z\ ?$  Hints will suffice, thank you. Needing to upgrade Squid 3.5.12 on Ubuntu 16.04.3 LTS to a more recently patched version (Squid 3.5.23) I found some instructions here:   Entered the first 3 commands exactly and got a note from apt-get: "N: Skipping acquire of configured file 'main/binary-i386/Packages' as repository ' xenial InRelease' doesn't support architecture 'i386'"  I don't have typos and nonexistent foreign architectures as others have done:  root@wprdsqd01:/# dpkg --print-foreign-architectures i386 root@wprdsqd01:/# dpkg --print-architecture amd64 root@wprdsqd01:/#   Does this just mean that the repository doesn't have i386 (32bit) binaries ?  Thanks Where can I find the points per game throughout many NBA seasons? Where can I find the points per game (of both teams) throughout many NBA seasons?  To illustrate my point, what I mean is something like:   2000/01 average points per NBA game: 200; 2001/02 average points per NBA game: 210; 2002/03 average points per NBA game: 205;   so on...