I am currently reading through a solution of the following problem and I am having a bit of trouble. The problem is as follows:     Let $S$ be a set and $F$ be the free group on $S$. Show that if $|S| &gt; 1$, then $Z(F)$ is trivial.    In order to deduce a contradiction, suppose that $w \in Z(F) \setminus \{e\}$ (where $e$ is the identity element). Let $w = a_{1}^{\pm1} \cdots a_{n}^{\pm 1}$ be the reduced word expression for $w$. Since $w \in Z(F)$, we have  $$ \left(a_{n-1}^{\mp 1} \cdots a_{1}^{\mp 1} \right)w = w\left(a_{n-1}^{\mp 1} \cdots a_{1}^{\mp 1} \right). $$ Now the left-hand side reduces to  $$ \left(a_{n-1}^{\mp 1} \cdots a_{1}^{\mp 1} \right)w = \left(a_{n-1}^{\mp 1} \cdots a_{1}^{\mp 1} \right)\left(a_{1}^{\pm1} \cdots a_{n}^{\pm 1} \right) = a_{n}^{ \pm 1}. $$ Comparing this to the right-hand side, we deduce that it must be the case that $a_{n}^{\pm 1}=a_{n-1}^{\pm 1} = \cdots = a_{1}^{\pm 1}$; so write $w = \left(a_{1}^{\pm 1}\right)^n$.    [Now every part of the proof makes sense to me thus far, but I do not see the rationale for the coming part.]     Since $|S| &gt; 1$, let $b \in S$ be such that $a_{1}^{\pm 1} \neq b$. Then $wb = bw$ because $w \in Z(F)$, and so $\left( a_{1}^{\pm1} \right)^n b = b\left(a_{1}^{\pm 1} \right)^n$. However this implies that $a_{1}^{\pm 1} = b$, which is a contradiction.    I cannot see how to deduce that $a_{1}^{\pm 1} = b$ from $\left( a_{1}^{\pm1} \right)^n b = b\left(a_{1}^{\pm 1} \right)^n$. The solution says to consider the reduced words, but this does not seem to be very helpful considering $\left( a_{1}^{\pm1} \right)^n b$ and $b\left( a_{1}^{\pm1} \right)^n$ are already reduced.
How can I show that "A center of a free group that is non-cyclic is trivial" ?
The repository 'http://dl.google.com/linux/chrome/deb stable Release' does not have a Release file After executing this command  sudo apt-get update &amp;&amp; sudo apt-get -y upgrade   I get this error  W: The repository 'http://dl.google.com/linux/chrome/deb stable Release' does not have a Release file. N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use. N: See apt-secure(8) manpage for repository creation and user configuration details.   I've seen past inquiries on same issue but the answers given are specific to other repositories.  Any help will be appreciated. What's better here - a single column or multicolumn primary key? Let's assume you have a table groups and a table item. Each item belongs to exactly one group. It is an inherent part of that group. An item cannot exist outside of a group and it cannot be moved to another group.  When trying to decide on a primary key for the item table, what should I use?  Should I make up an artificial global serial key like this:  CREATE TABLE items (     item serial PRIMARY KEY,     group integer NOT NULL REFERENCES groups(group), );   ... or should I rather use a composite key and per-group item serial like this:  CREATE TABLE items (     group integer NOT NULL REFERENCES groups(group),     item integer NOT NULL,      PRIMARY KEY(group, item) );   The reason why I'm leaning more towards the second solution is that the post URL will always show the group and item, so it makes sense to have both of them as the composite primary key. In case of the first solution, the URL contains superfluous information because the group ID can already be deduced from the item ID alone. The URL structure is given, however, and cannot be changed.  The disadvantage of the second solution is that you have to manage a per-group serial (i.e. each item integer should start from 0 for each group).  What's better in terms of best practices, normalization and performance? Or is it simply a matter of taste? Do I have a virus on my phone? A message from an Android app says so I have just received a message from an Android app on my phone saying my phone has been infected with a virus and to click a link to fix the problem.   I'm not sure I trust this, is it possible to have a virus on Android? Can anyone tell me if there is a way with generics to limit a generic type argument T to only:   Int16 Int32 Int64 UInt16 UInt32 UInt64   I'm aware of the where keyword, but can't find an interface for only these types,  Something like:  static bool IntegerFunction&lt;T&gt;(T value) where T : INumeric How does Ubuntu flavor iso differ from a desktop package? TL;DR: how does an Ubuntu flavor iso installation ( for example Lubuntu ) differ from installing a desktop package via apt-get ? (for instance sudo apt-get install lubuntu-desktop)  Context for the question:   Some time ago I answered a question on AskUbuntu , where I tested in virtual machine removing unity-desktop and installing ubuntu-gnome-desktop. That makes the boot logo ( plymouth ) change, desktop changes, different packages get installed (gnome-network-manager instead of network-manager , if I am not mistaken). It appeared that in effect I've switched to different Ubuntu flavor, even though I started out with the standard Ubuntu system. Hence the question: what differentiates the pre-made Ubuntu flavor iso from installing a package ? Is it all the same base underneath the hood or does installation from iso have specific configurations ? If it's possible to install a flavor via package manager, why then separate flavor iso's exist ? (obvious answer is user convenience, but I think there has to exist other reasons. Canonical is a business , and it makes not a lot of sense for a business to provide additional flavor iso , if a sysadmin or end-user can get the same thing via package manager by themselves). How can I get the Cartesian product (every possible combination of values) from a group of lists?  Input:   somelists = [    [1, 2, 3],    ['a', 'b'],    [4, 5] ]   Desired output:  [(1, 'a', 4), (1, 'a', 5), (1, 'b', 4), (1, 'b', 5), (2, 'a', 4), (2, 'a', 5) ...] How to execute consecutive commands from history? Suppose I want to execute a sequence of four commands that I have executed before. If the first one is 432 in the command-history, then I could do:  $ !432; !433; !434; !435   I'm curious, is there a more efficient way to accomplish this? 17.10 Super key to open applications When does a flying creature start and end falling? Situation:  Alice the Aarakokra sees Bob the Battlemaster coming. She ascends to 605', just out of the 600' range of Bob's longbow.  Bob (Str 14) performs a running high jump 5' into the air and successfully hits Alice with his longbow, despite firing at disadvantage for long range. He spends one of his maneuver dice to perform the Trip Attack maneuver, and succeeds on the contested roll to knock Alice prone.  Alice is now flying and prone, so she falls. Falling does 1d6 bludgeoning damage per 10' fallen, to a maximum of 20d6.  Questions:   When does Alice start falling? It is on Bob's turn as soon as she's knocked prone, or some time else such as the beginning of her own next turn? When does Alice hit the ground? On Bob's turn as soon as she's knocked prone, or some time else?   These answers determine whether anyone can save Alice from her 20d6 of certain doom.   Is a reaction (e.g. casting Feather Fall) on Bob's turn the only hope? Can Alice's ally Charlie swoop in on his subsequent turn within the same round to catch her? Can Alice spend half her movement on her next turn to "stand from prone" and pull out of her involuntary dive? If Alice had expected to be shot down, could she have spent an action to be Ready to "stand from prone" while moving through Bob's airspace? I'm using Lubuntu for a while now, but found a couple of details that I wanted to view with Unity (that's another story).  Ok, the thing is: I've tried on a VM with Lubuntu to install Unity (sudo apt-get install unity) When the installation is over. I selected in the Lubuntu login Unity/Unity2D but did not work.  Right now I'm installing ubuntu-desktop, I am presuming that will work, but also it will change my login for example.  Is there a way to do a "clean" installation of Unity but not make major changes on my Lubuntu?  I would prefer Gnome-Shell instead of Unity. How can I find the smallest set of groups of $n$ elements such that every element is in the same group as every other at least once? Background: I'm working on  for , and I've run into a problem with how I'm creating the individual matchups (groups of 4 entries).  Currently, I'm simply generating all combinations of 4 elements (in the combinatoric sense) of entries, but that gets really big really quickly:  $$x \choose 4$$    (from )  Therefore, my question is: How can I choose subsets (of length $n$) of a set such that each element of that set appears in the same subset as every other element at least once, while maintaining a minimal amount of subsets?  I can figure this out for $n = 2$, since the solution is quite intuitive: simply combine the first element with every other element, combine the second element with every other element that comes after it, etc. For a set of length $L$, this yields:  $$\sum_{x=1}^{L-1}(L - x)$$    (from )  which is much smaller and more manageable.  However, I still have not been able to figure out: How can I generalize this to any $n$? How can text selection be disabled in Android 3.0 Honeycomb? What's with the very slightly larger mass of the neutron compared to the proton? How can I  compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ? This question is updated to request additional answers and details in light of information now available in the Dungeon Master's Guide. In the released DM basic rules, Lost Mine of Phandelver, Hoard of the Dragon Queen, and the DMG teaser there are wands, staves, and rings that you can use to cast spells.  Do abilities that trigger from casting a spell activate when casting a spell from an item?  Can the character modify the spell using class abilities when casting a spell from an item?   Examples include:  An abjurer Wizard's arcane ward recovering hit points from a Dispel Magic cast from a Ring of Spell Storing  A sorcerer using Twinned Spell on a ray of enfeeblement from a Staff of Power  An evocation wizard using Sculpt Spell on a fireball from a Wand of Fireballs.   Now that the DMG is out we have some clearer rules on casting spells from items (DMG p. 141):  Some magic items allow you to cast a spell from the item.  The spell is cast at the lowest possible spell level, doesn't expend any spell slots, and requires no components, unless the item's description says otherwise  The spell uses its normal casting time, range and duration, and the user of the item must concentrate if the spell requires concentration. A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item.  Does whether or not you can apply class abilities to the casting of the spell depend whether the item uses the user's own spellcasting ability (such as the Staff of Fire) versus an item that comes with a fixed spellcasting ability built-in (such as the Wand of Fireballs)? Does it matter if it's something that triggers more or less automatically like Arcane Ward, versus something you have to control, like Careful Spell? Can a hotel kick you out if you let an unregistered guest share a room with you? We are going to an event and because of this event all hotels are fully booked. Now a friend wants to join us, but can't find accommodation. We are okay with him crashing in our hotel room, either on the couch or on a self-inflatable mattress. We have done this in the past where we secretly sneaked the "guest" in.   It is not that we don't want to pay the tourist tax, it is that when ask and the hotel says no, we can't sneak him in, because we drew attention to ourself. In many hotels you can just order another bed, but there are also a lot that don't have that service.   I don't have any ethical problems with my behavior of secretly hosting a guest. I did pay for the room. I don't like being sneaky about it, since again I do pay for the hotel room. Also if I have to choose between sneaking someone in the room to crash on the couch or let him/her sleep on bench in the park. I opt for the first.   Am I just worrying to much and will being honest about it in the end be acceptable to most if not all hotels? Does non-mass-energy generate a gravitational field? At a very basic level I know that gravity isn't generated by mass but rather the  and when I wave my hands a lot it seems like that implies that energy in $E^2 = (pc)^2 + (mc^2)^2$ is the source of gravity.  If the total energy of a particle contributes to the generation of a gravitational field, that seems to imply some strange things.  For example, you could accelerate a particle like an electron to such a high velocity that it has enough momentum / kinetic energy to collapse into a black hole.  An observer traveling with the electron would see an electron and stationary observers would see an arbitrarily massive black hole flying by.  These observations seem to be contradictory.  The same would be true of an extremely high energy photon.  Another consequence is that a stationary particle could appear to have so much mass that it could attract / deflect the trajectory of a rapidly moving observer.  It seems odd that at low velocities you wouldn't even notice a stationary electron as you pass it by but at high velocities the gravitational field you observe could be significant enough to alter your path.  Am I correct that non-mass-energy generates a gravitational field?  If it does, how are these strange observations like a particle looking like a black hole in one reference frame but not another reconciled? What does "\" mean in math Does equality of characteristic polynomials guarantee equivalence of matrices? I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and then the question.  True or False:   let $A,B \in \mathbb{Q}^{n \times n }$  Suppose $xI-A$ and $xI-B$ are equivalent then $\det(xI-A) = \det(xI-B)$.     I wrote in my notes this is true because any two matrices are equivalent in $\mathbb{Q[x]}^{n \times n}$ if and only if they have the same invariant factors.  Since the characteristic polynomial is a product of the invariant factors it follows that $\det(xI-A) = \det(xI-B)$.        True or False:   let $A,B \in \mathbb{Q}^{n \times n }$. If $\det(xI-A) = \det(xI-B)$ (in $\mathbb{Q}[x]$) then $xI-A$ and $xI-B$ are equivalent  in $\mathbb{Q}[x]^{n \times n}$   I think this is false because of the fact that equality of determinants is not strong enough to guarantee all invariant factors are equal but I do not have an example. Do I have to enter the UK on my UK passport? I'm a British National (Overseas), which basically means that while I don't have the right of abode in the UK, I am permitted to visit the UK without a visa. However, my passport has expired, and while it's being renewed, the schedule is a bit tight.  At the same time, I'm an US Citizen, and since I'm only planning to visit the UK, I don't need a visa if I enter on my US passport.  I'm not familiar with UK immigration law, and given that this probably isn't too common a situation, the UK Border Agency website doesn't have an answer that I can can find, nor do they seem to have a contact form that fits my situation, so here's hoping someone knows—given that I'm a British National (Overseas), am I required to enter the UK on my UK passport, or can I just use my (valid) US passport?