Can you check my proof that $H$ is characteristic? Suppose $|G|=pm$,where $p\nmid m$. Then if $H\mathrel{\unlhd}G$ and $|H|=p$, than $H$ has to be characteristic. We call a subgroup characteristic when $\varphi(T)\subset T$, $\forall \varphi \in Aut(G)$.  My proof: Suppose $g\in \varphi(H)$ and $g \notin H$. Since $\varphi$ is bijective $|\varphi(H)|=|H|=p$. So $\langle g\rangle|p$ i.e. $\langle g\rangle =p$. Since $H$ is a normal group and $\langle g\rangle$ is another subgroup, $\langle g\rangle H$ is also subgroup of $G$. So $p^2||G|=pm$ which contradicts to the fact that $p\nmid m$.  Is the idea right? Which parts would you advice to fix?
Normal subgroup of prime order is characteristic I'm stuck on this exercise from Herstein's topics in algebra:  Suppose that $G$ is a group and $|G| = pm$, where $p \nmid ~ m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.  How to prove that? What we know:   $H$ and $\varphi(H)$ have prime order $p$, so they must be cyclic and abelian They are both normal subgroups of $G$   how to show that, in fact, we have $H = \varphi(H)$ for every automorphism $\varphi$ ?
find command: -or option doesn't work I want to execute a command for an each file that matches a pattern. But the command that seems to be right, doesn't work for me, I have no idea why.  $ find . -type f -name '*.c' -or -name '*.h' -or -name '*.cpp' -exec ls {} \; $ ls script.sh  test.c  test.h How to connect to Salesforce REST API without OAuth? How does Jitterbit do it? Age limit is too strict Why does light of high frequency appear violet? Power Supply Capacity Formula There are many, many old posts on meta asking for suggestions on how to better organize favorites.  ,  and , among others.  This is not meant to be an open discussion on how to improve favorites, but rather a specific feature request that I'd like feedback on.      I would love to see a feature added that would let users arbitrarily create "groups" of favorites, and drag individual favorited questions under these groups.  I have about 2 pages of favorites (I know others have much more), and finding an individual one is quite difficult, especially since the specific question title is usually forgotten.  I just added  to my favorites, since I liked Andrew Whitaker's answer, and might want to come back and look at it later; I'd love to add this to a group called, say, "Useful Code".    I've favorited other posts since they had neat tricks in them, or because there were answers written on interesting theoretical computer science topics.  Having different groups for all these areas would be immensely helpful in finding posts later. Test condition in if command Any idea why echo statement is being executed in both scenarios?  Outputs it worked:  #!/bin/bash  rizwan='' if [ -n $rizwan ]; then   echo "it worked" fi   Replace -n with -z, and it outputs it worked when expecting no output:  #!/bin/bash  rizwan='' if [ -z $rizwan ]; then   echo "it worked" fi Diffinitive rules for using Android's getBaseContext, getApplicationContext or using an Activity's "this" I know that non-married lovers address each other by saying “This is my girlfriend/boyfriend.” I know that married couples address each other by calling “my wife/husband”.  I also know that gay lovers who are not married (yet?) address each other by saying "my boyfriend" (for gays) and "my girlfriend" (for lesbians).  But if they are married, how do they address each other?  And how do we describe them? I mean if they are gays, are they “husband and husband” or “husband and wife” (one of them acts like the wife)? And if they are lesbians, are they “wife and wife” or “husband and wife” (one of them acts as the husband)?  I know they can be referred to as a gay couple or gay lovers from , but I couldn't find the answer to address each of the two. Since  from other SE sites and is not thought to point out a lack of quality or helpfulness, but only to express agreement or disagreement, I'm wondering why downvoted answers here on meta are shown semi-transparent.  Shouldn't they be non-transparent then (like upvoted answers)?  Or am I missing something? How to trigger workflow from button in O365 I created a script editor webpart. I added a button which start a js function in the onClick event. I would like to start a workflow which is implemented on some list. Is this possible? Why does rubbing feet against sheets create a spark? Sometimes when I'm in bed at night, I rub my feet against the sheets and light is created. What exactly is going on?     I'm familiar with static electricity, and the triboelectric effect (creating a charge by rubbing 2 insulators together), and electrostatic discharge (releasing charge by touching a grounded conductor).     I believe that all of that is what is at work here, but I'm slightly confused since in other cases, the triboelectric effect and electrostatic discharge seem to occur as two separate steps, with different substances involved, and here, the two effects seem to be occurring simultaneously, with the same substances.     Can someone help me with a step-by-step explanation of the dynamics of what is going on? I am solving a clustering problem on a trivial dataset with the k-means algorithm. I am running a couple of tests and, from time to time, one centroid doesn't attract any points, i.e. I've got an empty cluster (see the purple "x" in the picture).  What should I do? Shall I delete it or just stop updating its value? Why?    I am aware that built-in functions (e.g., kmeans() in R) have automatic ways of dealing with this situation, but I am trying to write the standard algorithm from scratch. As soon as I fix it I'll be able to compare my results to built-in functions. At this moment I'm looking for some theoretical reasons why I should prefer one solution or another.  In the picture each colour represents a cluster according to the current iteration and each "X" is its centroid (old ones have been kept and marked with the number of the iteration in which they were computed).      [Some related links about empty cluster case, added later by @ttnphns: ; ; ; ; ; ; ; ; ] General trigger bulkification - best practices A general question asked from time to time on stack exchange is question of trigger bulkification.  Therefore I have 2 questions:   Which best practices are you using? What is the best way to handle updates of more than 200 records? (In case of more than 200 records the trigger is executed more than once.) Understanding central limit theorem What is wrong with the following sentence:      The Central Limit Theorem  implies that, as the sample size grows, the error distribution approaches normality.   Am I correct by saying that it should in stead state the MEAN of the sample error approaches zero as sample size grows? How Do I Compute the Eigenvalues of a Small Matrix? I understand that there are $\frac{8!}{3!2!}$=3600 permutations possible without any restrictions in place. I follow this by attaching all like elements as such {1,2,{3,3,3},{4,4},5} where I can account for 5! permutations where I have three 3s and two 4s in consecutive positions. However, when it comes to accounting for the permutations that, for example, will have three consecutives 3s but not so for the 4s, I am at a loss. I wrote some simple code and came up with a set of 960 permutations as the acceptable answer. As the question implies, I am unsure of how to reach that through other means. My apologies if this is considered elementary and I appreciate any help whatsoever. I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.  I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.  Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.   Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?  Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.  Suggestions about important theorems and concepts to learn, and book references, will be most helpful. What happens if I combine Ur-priest and prestige class to exceed 10 effective spellcasting levels? Suppose I have a human character who invested 5 levels on Wizard, 2 on Ur-priest(by appropriate feat chain), and 8 levels of Mystic Theurge, net level of 15. This allows you to cast spells as stated in 10th level Ur-priest Spells per day table. Now, assume that I leveled up, and I choose Mystic Theurge as the next level class. This makes me to look up Ur-priest table for Level 11, which does not exist. In this case, what happens to my divine spells per day? Does it continue to progress "in the same pattern", as Hellfire Blast progresses with Hellfire Warlock-Legacy Champion build? Does it stop and I only get Wizard spells per day? Or, isn't it allowed to do so?  I am totally aware that although the answer is "yes", it does not mean it means spells over 9th level will be granted to my character unless I take Improved Spell Capacity, which requires me to be epic. What I am curious is the progression of spells per day of existing spells.  P. S. Don't ask me how to satisfy the requirements. This is possible by combining Apprentice (Criminal), Iron Will and Great Fortitude as level-up feat, and Spell Focus (evil) as a Wizard bonus feat (RAW, it is a "Spell Focus" feat, anyway.) Novel(s) based on humanity fleeing from Earth and regressing to a pre-industrial state on a Colony so as to escape detection Shortly before his death, my father spoke to me about a series of novels that he had just finished reading which had a fascinating concept.  However I soon forgot the title(s) and author and I've been absolutely desperate to find out what they were.  The premise (as I understood) was that humanity had expanded out from the solar system into the local star systems before coming into contact with an incredibly aggressive species which was focused on wiping out the entire species.  Earth and her colonies were lost as remnants of the species fled.  Eventually it was decided by the military leadership to set up a colony in order to rebuild the human race.  However, in order for this colony to escape detection and to keep humanity from going back out into space and thus potentially running into this hostile alien race again, a series of satellites were set up around the planet chosen that would lock onto and destroy any form of technology (and usually the surrounding city) that went beyond pre-industrial standards.    One of the series protagonist is a long living android who possesses the consciousness of a human who was present at the original founding of the colony (and who seems shielded from detection) which observes as civilization is repeatedly regrown and torn down again and who attempts to try and reverse humanities self imposed lot in life.      From what I understood this was a series of novels, but I could be wrong. I've been trying to find this for years on my own with only a rough plot line as a guide and it's gotten me nowhere.