I am reading Michael Aschbacher's Finite group theory textbook. Exercire 1.6 makes me confused.   1.6 Let H is normal subgroup in G. Then the map L->L/H is a bijection between the set of all subgroups of G containing H and the set of all subgroups of G/H. Normal subgroups correspond to normal subgroups under this bijection.  It is not clear for me what is L. I perused previous paragraphs but didn't find any relevnt information. I will be really thankful if someone would help me to understandhow to solve this exercise.
I had studied group theory a year ago, but still could not understand the proof involving The Correspondence theorem.  let $G$ be a group and let $N⊴G$, where  $N⊴G$ indicates that $N$ is a normal subgroup of $G$. Then there is a bijection from the set of subgroups $A$ of $G$ that contain $N$ onto the set of subgroups $\overline A $=$A/N$ of $G/N.$  Can anyone explain the steps to me in:  $(1.)$ a very easy-going manner so that I understand importance of each step. $(2.)$I don't require the proof but only how the step follow each other .  I'll be thankful if there is anyone kind enough to do this....
Command line command to auto-kill a command after a certain amount of time I'd like to automatically kill a command after a certain amount of time.  I have in mind an interface like this:  % constrain 300 ./foo args   Which would run "./foo" with "args" but automatically kill it if it's still running after 5 minutes.  It might be useful to generalize the idea to other constraints, such as autokilling a process if it uses too much memory.  Are there any existing tools that do that, or has anyone written such a thing?  ADDED: Jonathan's solution is precisely what I had in mind and it works like a charm on linux, but I can't get it to work on Mac OSX.  I got rid of the SIGRTMIN which lets it compile fine, but the signal just doesn't get sent to the child process.  Anyone know how to make this work on Mac?  [Added: Note that an update is available from Jonathan that works on Mac and elsewhere.] What are the options for player characters to create traps? How do I recover my Google account (or Gmail) password or username? Is there any advantage in marketing my product for £99.99 instead of £100? Stop asking me if Community auto-flags were helpful I have  $m\times n = n + n + \dots + n$    in my text right now, but I would like to put a curly brace under the \dots to emphasize that there are m,n's involved. How would I do this? I am trying to think of what the name of the word/phrase is for when: An important task (not a difficult one, but a mission critical one) is at hand and instead of one person being assigned to it, a group takes on the responsibility to mitigate any consequences happening to one person in case something goes wrong and also to prevent a mistake or malicious intent occurring from one persons doing. (NOT because the task is big and needs to be divided in a group.) E.g.   Two different people needing to turn a key each in a device simultaneously to launch a missile.  Group of people tallying up votes for an election in case one of them were bridged to rig the outcome. Shell Type:  &gt;echo $SHELL /bin/ksh   Cronjob :   15 * * * * /bin/ksh /wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh &gt; /wls_domains/resMGT/logs/bea/data/script.log  The script is as below:  ##log files and lookup file "wlr3queue.txt" are in the same path of the script ##/wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh  Script :  #!/bin/ksh  dt=`date +%Y-%m-%d` xy=`date|awk '{print $4}'|awk -F":" '{print $1}'` yz=`expr $xy - 1` if [ $yz -le 9 ] then yz=0$yz fi  if [ $xy -gt 0 ] then  for i in `cat wlr3queue.txt` do  sum=0  count=`gzgrep -i "$dt" admin_resMGT_access.log* | grep "$yz:[0-5][0-9]" | grep "$i" | wc -l`  if [ $count -gt 0 ] then name=`echo $i | cut -d'/' -f3`  gzgrep -i "$dt" admin_resMGT_access.log* | grep "$yz:[0-5][0-9]" | grep "$i" | awk '{print $8}' &gt; consumed_time.txt  for j in `cat consumed_time.txt` do j_ms=`echo "$j * 1000" | bc` echo $j_ms &gt;&gt; consumed_ms_time.txt done  for k in `cat consumed_ms_time.txt` do sum=`echo "$sum + $k" | bc` done  avg=`echo "scale=2; ($sum/$count) "| bc`  min=`cat consumed_ms_time.txt | sort -n | head -1`  max=`cat consumed_ms_time.txt | sort -n | tail -1`  else  avg=0  min=0  max=0  fi ( echo $yz","$count","$avg","$min","$max )&gt;&gt;/wls_domains/resMGT/logs/bea/data/$name$dt.csv   rm -f consumed_ms_time.txt  done  else  echo "script won't execute at this hour" &gt; temp.txt  fi   I have executed the following commands and the script ran successfully.  ./wlr3queuetransaction.sh sh -x wlr3queuetransaction.sh /bin/ksh /wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh  /bin/ksh/ wlr3queuetransaction.sh.   How to debug? What to do? How can one show that a möbius transformation maps the unit circle to some axis? Suppose I have the möbius transformation f(z) = $\frac{1+z}{1-z}$ and I want to show that it will map the unit circle (excluding the point 1) to the imaginary axis.  Would it help to express my unit circle in polar form?     Here's what I did: f(e$^{i\theta}$) = $\frac{1+e^{i\theta}}{1-e^{i\theta}}$  But I'm not sure where to go from this step. Missing argument 2 for wpdb::prepare() Issue When thinking about travelling, I often have to think about a place to visit, and then look up visa rules for that particular country. This can be a fairly painful process because official government sites can often be hard to find, and advice on (say) guidebooks or Wikitravel tends to be generic.  Is there any resource where I can enter my nationality/residency, and then look up a list of visa rules for various countries? Let's say you have a few expressions like the following:  $$\begin{array}((x+17)^2 \\ x^2 + 34x + 289 \end{array} \\ 288 + \frac{x^2}{2} + \frac{x^2}{2} + 34x + 1 \\ [...] $$  You get the idea: there's an infinite number of ways of representing the same expression. The challenge is that my application is validating answers to math problems, input by math students practicing a certain concept. I'd like to make sure that whenever they submit an answer, the checker is completely insensitive to the issues of ordering, simplification, formatting etc. which would be plain frustrating. I'd also ideally like to support more than one variable such as x, y, z in the same expression.  What's a reliable way to check that two expressions are the same? The most brute force way I can think of would be to just plug a number into each variable of each expression and see what the result is.  I have a hunch however that there might be ways of getting false positives this way. That, and I might be either missing something really obvious about this, or there might be a better way of doing the comparison.  Would love to hear what you think. Ubuntu in VirtualBox wants to install on every boot I have installed Ubuntu 11.10 using Virtual Box in my Windows XP. I have use Ubuntu 11.10 iso to install it. First time when I installed every thing was fine but when I shutdown the VirtualBox and start Ubuntu again, it asks me to install Ubuntu and try Ubuntu without installing.  Do I need to do some extra settings? Dilemma: Present Perfect Or Perfect Continuous? Did spacetime start with the Big Bang?  I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence/existence of any other space-time before the big bang? Can a well-timed Twisted Image kill a player using Tree of Redemption? What happens if my opponent activates  and I cast  with the life-exchange ability on the stack?  Looking at the comp rules, these seem most relevant:     701.8a A spell or ability may instruct players to exchange something (for example, life totals or    control of two permanents) as part of   its resolution. When such a spell or ability resolves, if the   entire exchange can’t be completed, no part of the exchange occurs.      Example: If a spell attempts to exchange control of two target creatures but one of those              creatures is destroyed before the spell resolves, the spell does nothing to the other              creature.      112.7a Once activated or triggered, an ability exists on the stack independently of its source.    Destruction or removal of the source   after that time won’t affect the ability. Note that some    abilities   cause a source to do something (for example, “Prodigal Sorcerer deals   1 damage to    target creature or player”) rather than the ability   doing anything directly. In these cases, any    activated or triggered   ability that references information about the source because the   effect    needs to be divided checks that information when the ability   is put onto the stack. Otherwise, it    will check that information   when it resolves. In both instances, if the source is no longer in the   zone it’s expected to be in at that time, its last known information   is used. The source can still    perform the action even though it no   longer exists.   Is the entire exchange ability countered because the Tree is dead by the time it resolves? Or do we use the "last known state" (toughness 0), thereby killing its controller? What are good methodologies for creating sprocs that reduce the pain of debugging? And what tools are out there for debugging stored procedures?  Perhaps most importantly, what are indications to look out for that errors are happening in a sproc and not in the code? I hope I'm not all over the board too terribly bad here. Votes for answers to any of the above. Thanks.  For what it's worth, I work in a .NET environment, SQL servers. What's the equivalent of the "cls" command from Windows/DOS? Is there any shortcut/easy way to switch this system proxy settings? Is there any module or some configuration that I can define the meta tags for the views page display in drupal 7 using  3.0?