I tried it using proof by contradiction:  Suppose that there are even number of elements of order $2$. Call them $x_1, ..., x_{2m}$, where $x_1,...,x_{2m} \neq e$. Then, consider the set $G=G \setminus \{e,x_1,...,x_n\}$. The size of $G'$ is odd since the size of $G$ is even.  Elements in G' does not have itself as an inverse, since $x^2=e \iff x=x^{-1}$.  Also, note that each elements in G' has a different inverses. (Suppose $a\neq b$, $\ a,b \in G'$, and $\exists c \in G'$ such that $ac=bc=e$. Then, $a=b=c^{-1}$ which is a contradiction.)  This shows that $|G'| =$ even which is a contradiction.    This seems complicated for no reason. Is this correct? and how do I make my proof neater?
I am working on the following problem from group theory:     If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd.   That is, for some integer $k$, there are $2k+1$ elements $a$ such that $a \in G,\;\; a*a = e$, where $e$ is the identity element of $G$.
Why do students go for a PhD even when chances of getting academic jobs on its completion are slim? This is in reference to my question: .  I was told in the answer of the above linked question that chances of landing a job as an assistant professor in Mathematics are quite slim. Very few people will actually get a permanent position in academia.  An article of 2014 shows only 0.5-3% of PhD graduates will actually get a job as an Assistant Professor.  But I have two questions:        Why do so many people still apply for a PhD in some reputed institutes in India even knowing the harsh reality? I found there are 1500 applications for four PhD scholarship positions.          One answer says that people move to industry, but I still don’t understand how after doing a PhD in suppose differential geometry or algebraic geometry one can do a job in industry if he/she gets no position as an assistant professor.   Will one be satisfied to do some managerial work in some Consultancy firm after doing a PhD in Mathematics?      Are people jobless after a PhD?   Completion of a PhD requires so much effort. So why is there no value after doing it?      NOTE:  In the most up-voted answer below it has been said that many who enroll for  a PhD don't even wish to pursue an academic career.  I have been asking my seniors who are currently pursuing a PhD or are in the verge of completing their degree or who have completed it. They said that they all want to be Professors in some Universities and want to be involved in research because that is the dream job for them.  I am obviously talking about PhD in Mathematics How to watch live youtube stream using vlc? Is it possible for now to watch live youtube video streams using vlc? I just don't want to use flash player. I know that vlc can play youtube videos using link for the page where that video is located, but how about live youtube videos? What really causes light/photons to appear slower in media? Get the subdomain from a URL Getting the subdomain from a URL sounds easy at first.  http://www.domain.example   Scan for the first period then return whatever came after the "http://" ...  Then you remember  http://super.duper.domain.example   Oh. So then you think, okay, find the last period, go back a word and get everything before!   Then you remember  http://super.duper.domain.co.uk   And you're back to square one. Anyone have any great ideas besides storing a list of all TLDs? Is it possible to prevent LUDecomposition from pivoting? Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$. How to enable or disable a user? I'm uing ubuntu 12.04 desktop. I have 3 users: user1(administrator), user2(standard) and guest. I wanted to disable user1 and enable user2 which auto logs on with no password but after I did that I can't login to user1 and user2 accounts except the guest session user. I'm striped off every administrative privileges. I don't know which options are available to me and how do I enable root or user which is an administrator? Suppose $f,g$: $\Bbb R$ → $\Bbb R$ are continuous functions such that $f(r) = g(r)$ for all $r$ in $\Bbb Q$. That is to say, $f$ and $g$ are equal on the rational numbers.   Prove that  $f(x) = g(x)$  for all $x$ in $\Bbb R$ Collinearity diagnostics problematic only when the interaction term is included How can I make my touchpad scroll either vertically or horizontally, but not both? If I'm browsing the internet and a page has both horizontal and vertical scroll bars it becomes really irritating to scroll with my touchpad. It keeps scrolling horizontally as well as vertically!  I'm using Ubuntu 16.04  and below are touchpad settings  Device 'SynPS/2 Synaptics TouchPad':     Device Enabled (160):   1     Coordinate Transformation Matrix (162): 1.000000, 0.000000, 0.000000, 0.000000, 1.000000, 0.000000, 0.000000, 0.000000, 1.000000     Device Accel Profile (282): 1     Device Accel Constant Deceleration (283):   2.500000     Device Accel Adaptive Deceleration (284):   1.000000     Device Accel Velocity Scaling (285):    12.500000     Synaptics Edges (286):  1769, 5431, 1652, 4658     Synaptics Finger (287): 25, 30, 0     Synaptics Tap Time (288):   180     Synaptics Tap Move (289):   242     Synaptics Tap Durations (290):  180, 100, 100     Synaptics ClickPad (291):   0     Synaptics Middle Button Timeout (292):  75     Synaptics Two-Finger Pressure (293):    282     Synaptics Two-Finger Width (294):   7     Synaptics Scrolling Distance (295): 200, 200     Synaptics Edge Scrolling (296): 0, 0, 0     Synaptics Two-Finger Scrolling (297):   1, 1     Synaptics Move Speed (298): 1.000000, 1.750000, 0.036300, 0.000000     Synaptics Off (299):    1     Synaptics Locked Drags (300):   0     Synaptics Locked Drags Timeout (301):   5000     Synaptics Tap Action (302): 2, 0, 0, 0, 1, 3, 0     Synaptics Click Action (303):   1, 1, 0     Synaptics Circular Scrolling (304): 0     Synaptics Circular Scrolling Distance (305):    0.100007     Synaptics Circular Scrolling Trigger (306): 0     Synaptics Circular Pad (307):   0     Synaptics Palm Detection (308): 0     Synaptics Palm Dimensions (309):    10, 200     Synaptics Coasting Speed (310): 21.120001, 50.000000     Synaptics Pressure Motion (311):    30, 160     Synaptics Pressure Motion Factor (312): 1.000000, 1.000000     Synaptics Resolution Detect (313):  1     Synaptics Grab Event Device (314):  0     Synaptics Gestures (315):   1     Synaptics Capabilities (316):   1, 0, 1, 1, 1, 1, 1     Synaptics Pad Resolution (317): 83, 46     Synaptics Area (318):   0, 0, 0, 0     Synaptics Noise Cancellation (319): 8, 8     Device Product ID (277):    2, 7     Device Node (278):  "/dev/input/event6" The error message is :the grub-efi-amd64-signed package failed to install into /target/. Without the GRUB bootloader the system will not boot  I'm trying to install on HP pavilion g6-2303 TX using USB.  How do I fix this problem? I'm thinking about buying the new MacBook Pro with 4 USB-C ports.  I want to attach two screens, mouse and keyboard and ethernet to the MacBook Pro while using the fewest number of USB-C ports possible.  So what I need is an adaptor with:   two DP or HDMI ports (or one DP port with daisy chaining support) two or more USB ports an ethernet port   Best case would be a hub to which all devices are connected with a single connection to the MacBook Pro, including power supply. Worst case if attaching the MacBook Pro to the devices would require me to use all four of USB-C ports.  Given the answers up to far, I can tell that:   Daisy chaining might or might not work with the MacBook Pro (since it requires DP1.2 which is supported by the graphics chipset, but it's uncertain if it's supported by the MacBook Pro itself) I can attach the monitors with two cables as a last resort, leaving me with two ports for the remaining peripherals.   What I still don't know:   Would daisy chaining work with a hub or does it require the first monitor to be directly attached to the MacBook Pro? Does some kind of hub exist supporting attachment of all listed devices with a single upstream connection to the MacBook Pro? Proving replacement theorem? I want to see if I am understanding the proof of the replacement theorem correctly.   Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors.  Let $L \subseteq V$ be a linearly independent subset containing $m$ vectors.  Then $m\leq n$ and there exists a subset H of G containing exactly $n-m$ vectors s.t. $L \cup H$ generates $V$.   The proof in the book says to start with $m=0$. In that case $L=\varnothing$, the empty set.        Let's assume this theorem is true for some integer $m \geq 0$.   Let $L = \{v_j\}_{j=1}^{m+1}$ and define it as a linearly independent subset of $V$ consisting of $m+1$ vectors.  Since any subset of a linearly independent set is linearly independent as well ($S_1 \subseteq S_2 \subseteq V$), then $\{v_j\}_{j=1}^m$ is linearly independent also.   It then says to use the induction hypothesis to say that $m \geq n$. Well, ok, but that seems sort of pulled out of nowhere to me. The next step is to say that there is therefore another subset, $\{u_k\}_{k=1}^{n-m}$ of G such that $\{v_j\}_{j=1}^m \cup \{u_k\}_{k=1}^{n-m}$ spans $V$.   That being the case there are scalars $(a_j)_{j=1}^m$ and $(b_k)_{k=1}^{n-m}$ which you can multiply by the vectors $v_j$ and $u_k$. You can then add the two sets of vectors, yielding $$\sum_{j=1}^m a_j v_j + \sum_{k=1}^{n-m} b_k u_k = v_{m+1}$$  At this point I am not sure I understand things. Because we are assuming $n-m&gt;0$ -- otherwise $v_{m+1}$ is linearly dependent. But then it says not only is $n&gt;m $ but $n&gt;m+1$, and that is where I am losing the plot to get to the next step. So if I'm correct, when you chose your starter gear and wealth you can either choose to roll some dice depending on your class and buy everything from that or you can choose the starter gear from your class and add the background items (including some gold). Here are my 2 questions:   If I choose the gear from my class, do I have to take all the equipment from only one of the (a) or (b) choices, or can I freely mix-and-match choices from the two sets? Can you spend the extra gold you get from your background on extra equipment (e.g. some soap and some fishing equipment) or do you have to start with that gold in its pouch?   The reason I ask is that I have recently created a level 1 wood elf druid and I mixed-and-matched the gear from the class, so that I chose a shortbow as a "(b) simple weapon" for the first point and a "(a) Scimitar" at the second point. My DM wasn't so happy about this, as he can't understand why I'm mixing (a) and (b).  He also wasn't so happy with me spending my 10gp from my outlander background. I decided that my druid wouldn't lay so much worth in 10gp compared to a bunch of stuff I got for that (e.g. some soap and some fishing equipment and some other useful things) and he seems to think that I'm supposed to just start with the gold instead. How do injuries work with Polyjuice Potion? We know that if you (using Polyjuice), take the form of someone who had been injured, you assume that injured form (e.g. Barty/Moody). If you are injured, presumably you would be uninjured while under Polyjuice. My question is, is this in real-time? That is, if you take a Polyjuice, and while you're still under its effects, the person being transformed into gets injured (or even dies), will that injury be transferred onto you? And conversely, if you get injured while in someone else's body, will that transfer over to your real body after the Polyjuice wears off, will it disappear, or will it go to the person you turned into? Why does the Moon not revolve around the Sun directly? The sun pulls on the moon with a force that is more than twice the magnitude of the force with which the earth attracts the moon. Why, then, doesn’t the sun take the moon away from the earth? How to partition for Windows/Ubuntu dual boot on same SSD I have a 500GB SSD in which i would like to have both Windows and Ubuntu. My plan is to have something similar to this: Windows 8.1 and ubuntu 14.04 Uefi boot   an efi partition (created by windows installer) which will have Grub as bootloader 150GB ntfs for Windows 15GB ext4 for Ubuntu (root) 35GB ext4 for Ubuntu (home) ~8-16GB swap for Ubuntu (I have 16GB RAM) ~284-292GB ntfs for Data shared for both Windows and Ubuntu   My current setup is   200GB ntfs Windows 300GB ntfs for Data   I plan to shrink both the Windows and the data partiton in order to achieve my desired plan.  My questions:   Does the order of partitions matter? Does it matter if the ubuntu partitions are not at the end/beggining of the drive? Does the swap partition behaves better if its the last/first partition? Can I use the shared NTFS Data partition between windows and ubuntu without encountering problems? Do I have enough space for my ubuntu /home partition (thinking that I'll have access to the ntfs shared partition)?   Edit: Laptop, Asus N56jr, i7, 16Gb Ram, SSD samsung 850 evo 500gb  Windows 8.1 and Ubuntu 14.04 Using \left( & \right) around amsmath's align delimiter ("&") I want to align equations (using the align environment from the amsmath package) inside matching \left( and \right) commands. For example:  \documentclass[a4paper,12pt]{article}  \usepackage{amsmath}  \begin{document}  \begin{align}     a \left( b &amp;= c \right)\\     d \left( e &amp;= f \right) \end{align}  \end{document}   But that code gives this error:     Extra }, or forgotten \right.   The error disappears when I remove the delimiters. Is there a workaround for this? I am trying to calculate the force of impact of a falling object. I did my egg drop project. I dropped the egg from 10m with a mass of 126kg and with the velocity of 14.1 m/s. Which formula should I use to calculate the force of impact? I found the kinetic energy first and later I used the following formula: $W=Fs$ to find the force of impact. Is that the right formula? Push Faces Like Sketchup Is there a way in Blender to push faces like Sketchup does?     The best I can get is doing a regular extrude and a lot of work to clean the "mess" up...    Thank you for your time!