Let $(G,*,{}^{-1},e)$ be a group and $a\in G$ fixed element. We define a new operation $\circ$ on $G:$ $$x\circ y=x*a*y.$$  I have to show that $(G,\circ)$ is, also, a group. I don't know how to find inverse element of a element $x.$ I get that neutral in this structure is fixed element $a.$ Then  $$x\circ x^{-1}=a$$ and $x*a*x^{-1}=a.$ What to do next?
Let $G$ be a group and $c$ be a fixed element of $G$. Define a new operation $*$ on G by:   $$x*y=xdy$$   for all $x$ and $y$ in $G$, where $d$ is the inverse of $c$. Prove that $G$ is a group under the operation $*$.    If you want to check, this problem is in the book “A Course In Group Theory”, by John F. Humphreys (problem $3$, page $28$)   I know what are the axioms that $(G,*)$ must obey in order to be a group and I did show that all the axioms where obeyed except the inverse axiom.  My problem is that, whenever I try to verify what would the inverse be for some element $x$, I get an equation where the $x$ is cancelled in both sides.  Can please someone guide me?  Thank you!  (P.S.: This is NOT a homework assignment, I’m studying Group Theory by this book and got problems solving this question)
What is the behaviour of \fontsize if baselineskip is below a certain value? during my optimization process I found something spurious I cannot explain to myself and which I would like to understand. So with \fontsize you set the fontsize and the baselineskip. However the behaviour of the second paramater is a bit unclear to me, because if you reduce it too much, at a certain point the baselineskip gets bigger again. Let me show you my example: \documentclass[landscape]{article} \usepackage[margin=1cm]{geometry} \usepackage{lipsum,multicol}  \begin{document} \begin{multicols}{4}     \fontsize{10}{11}\selectfont     \lipsum[1]     \columnbreak          \fontsize{10}{10}\selectfont     \lipsum[1]     \columnbreak          \fontsize{10}{9}\selectfont     \lipsum[1]     \columnbreak          \fontsize{10}{8}\selectfont     \lipsum[1]     \columnbreak \end{multicols} \end{document}  If you compile this, your result will be looking like this:  Now from the first to the third column it seems logic, that the baselineskip is reduced and hence less space is needed vertically to typeset. But with the third column, even though the baselineskip is reduced even further, it needs more vertical space overall. It is strange however, that for example from line 11 to 12 (starting with tus and lum), the spacing seems to be narrower than in column 3, while in a lot of line the spacing is bigger. Changing the baselineskip to lower values down to 0 does not modify the vertical alignment further. What is the purpose of this behaviour and is it intended? I cannot see a benefit of this and I am a bit confused. I want to write as a math fraction 1/4 " ( a quarter of inch). I want to write this fraction out of an equation because is a characteristic of an element. How can I hover over icons in Ubuntu Dock to see all windows of one application? I'm using Ubuntu 17.10 with GNOME 3. How can I show all open windows of a certain application using the sidebar (aka Ubuntu Dock)? Right now I have to right-click and select "all windows" to see my other open windows.   Is there a quicker way to do this? Word for Negatively Surprising I'm searching for a word representing "Negatively surprising", or "Negatively amused" in an appropriate form. Example: Someone who proclaims he aims to do things in the right way has done something in the completely opposite way.    How could you have done such a thing? That action is very ___.   Said with the purpose of showing the contrast between the expected and the actual, and asking for change to adhere with his set goals, instead of just pointing out disappointment.  It is different from  because it asks there about surprise of being affected by the disappointment. The meaning of surprise here is different. "None" as plural indefinite pronoun In my grammar book (English Grammar, HarperCollins Publishers), I read that none is occasionally treated as plural, but it is usually regarded as singular. Can you give me an example of sentence where none is used as plural pronoun? Why is "what can I ask about here" not the first entry in the help center? On a lot of sites, the first section of the Help Center looks like this:    When I go to a site's help center it's almost always to find out what's on-topic there, because I'm considering asking or migrating a question.  Leading with what not to ask seems odd to me, and I suspect that to someone not already familiar with SE it could be a little off-putting.  It's not always like this, though; here on MSE we get this, which I've seen on some other sites too:    That ordering makes a lot more sense to me.  It's also consistent with the tour.  (Maybe "how do I ask a good question?" should also be "above the fold", in the top five somewhere.)  Why the difference?  Can we either make the order consistent across the network and more logical, or give users or moderators the ability to change the order on their sites? How to smooth shade an object while retaining hard edges? Can Recovery Mode install OS X onto a blank SSD? I want to install a new Solid State Drive (SSD) into the Mac. Can I just build in the SSD, start the MacBook in Recovery Mode and install OS X from there on?  After reading a lot on the internet I still have some doubts…  My HDD crashed, I can't repair it using disk utility or the fsck methods as mentioned on the internet.  According to  it should be possible.  I know how to replace the HDD with the SSD. I would go for the Samsung 850 EVO SSD - 500GB. My Mac is a MacBook Pro Mid 2012, model A1278. Is is possible to move all windows (or all not minimized windows) form one workspace to another?   I know I can move one window to other workspace with Shift+Ctrl+Alt+arrow, but it will move only that one focused window. What really allows airplanes to fly? Vim-latex adds extra brackets when typing empty brackets What's "Another Way In"? There's a new achievement and I'm stumped as to how to get it:     Start a game without breaking the laboratory wall   I assume it has something to do with the new Gadgets but none of them sound like they'd help. How do I get this achievement? Please check my simplification for x? Is the Nintendo 3DS region-locked? I am thinking of buying a 3DS when the price drop comes into place in America, however I live in Australia. Will Australian 3DS games work on a North American 3DS? Hiding downvoted questions silences important discussions on Meta I just noticed a particular MSE post today on a topic I care about that was posted almost a week ago, which I entirely missed before. I'm a pretty avid MSE user, and this happens quite rarely to me.     was created by an SE employee, and is essentially the final answer to , which is the 10th most upvoted MSE post with a bit more than 900 upvotes right now. So it's kind of an important post, and on a topic I care about, but I still missed it entirely until it was linked in a comment somewhere else.  The reason I missed it is because it's heavily downvoted. This means that as it has below -8 score, it's hidden from the home page entirely, drastically reducing the visibility of the post. In particular, the negative score removes all visibility a question usually receives from being bumped due to edits, new answers and bounties.   Many of the recent heavily-downvoted Meta posts by SE have been featured manually, which grants them a lot of visibility beyond the particular meta site. This masked the problem pretty efficiently, but it is more of a workaround, and one that isn't applicable to all situations.  Voting means something different on meta sites, and not every downvoted question should be hidden from view. Closed downvoted questions can certainly still be affected by the usual rules on the home page, and that should cover the vast majority of questions that actually have a purpose being on the home page. But open questions should be displayed on the home page on meta sites even if they're heavily downvoted. How do I set environment variables? I'm trying to set up Apache Tomcat on my pc, and it wants me to set up an environment variable for CATALINA_HOME.  Does any know how to do this? Expected value of a non-negative random variable How do I clean the inside of my viewfinder? My viewfinder (not LCD) has spots of dust on it.  What is the best way to clean the inside?  I have cleaned the outside with a QTip. Whenever I create a document in Microsoft Word, it complains about a lot of my sentences being in passive voice. But, when I read that sentence aloud, it sounds fine to me. I am not sure if it is just me and will a statement in passive voice sound strange to a native speaker?  So, my question is, is it considered bad form to use passive voice generally? Or in some specific cases like written communications only?  Edit: If it is ok to use passive voice, then why does MS-Word complain? Labels disappear while zooming