The intersection of maximal subgroups of abelian group is not trivial. I want to find the sufficient and necessary condition of intersection of maximal subgroup of abelian group is not trivial. I showed intersection of maximal subgroup of a group $G$ is normal subgroup of $G$. I know fundamental theorem of abelian group. Thank you for your help.
Frattini subgroup $Φ ( G )$ is not trivial Frattini subgroup $Φ ( G )$ of a group G is the intersection of all maximal subgroups of $G$. The question is  Let $G$ be an abelian group.What is the sufficient and necessary condition of $Φ ( G ) ≠｛e｝$？  I think if $G$ has no torsion part, we cannot say $Φ ( G ) ≠｛e｝$,because $\Bbb Z ×\Bbb Z×・・・ ×\Bbb Z$ has a maximal subgroup $p\Bbb Z ×\Bbb Z×・・・×\Bbb Z$, with intersectioning distinct primes, we gain just a trivial group. So, we need torsion part. But I cannot proceed from here.I only know the answer, $G$ has an element of order $p^2$, that  is, $G$ has theree subgroups. I want to deduce this consequence from steady discussion. P.S  According to this answer, in a ring that is a finite direct product of quotient rings of $\mathbf{Z}$, we can say rattini subgroup is the same as the Jacobson radical of the corresponding ring.
Delete from uitableview + write to plist not working There are several reasons to be looking for a ROM:   Upgrading or downgrading your stock ROM Looking for firmware from a different region/language Flashing a custom ROM for different features, removing bloat, etc.   But where can one find ROMs in the first place?  What sites are likely to have the ROM I need?  ; please add to the wiki answer instead of posting separate answers.  See also: Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$ If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$? I am trying to show that: \begin{equation} \frac{n}{\phi(n)} = \sum_{d \mid n} \frac{\mu^2(d)}{\phi(d)} \end{equation} where $\phi(n)$ is  and $\mu$ is the .  The identity clearly holds for $n=1$, so if we write $n=p_1^{a_1} \cdots p_k^{a_k}$ for the prime decomposition of $n$, the left-hand side becomes: \begin{eqnarray*} \frac{n}{\phi(n)} &amp; = &amp; \frac{p_1^{a_1} \cdots p_k^{a_k}}{\phi(p_1^{a_1}) \cdots \phi(p_k^{a_k})} \\ &amp; = &amp; \frac{p_1^{a_1} \cdots p_k^{a_k}}{(p_1^{a_1-1})(p_1-1) \cdots (p_k^{a_k})(p_k-1)} \\ &amp; = &amp; \frac{p_1 \cdots p_k}{(p_1-1)\cdots (p_k-1)} \end{eqnarray*}   Whereas the right-hand side becomes:  \begin{eqnarray*} \sum_{d \mid n} \frac{\mu^2(d)}{\phi(d)} &amp; = &amp; \sum_{\substack{d \mid n \\ p^2 \nmid d}} \frac{1}{\phi(d)} \\ &amp; = &amp; 1 + \frac{1}{\phi(p_1)} + \dots + \frac{1}{\phi(p_k)} + \frac{1}{\phi(p_1 p_2)} + \dots + \frac{1}{\phi(p_{k-1}p_k)} \\ &amp; + &amp; \dots + \frac{1}{\phi(p_1 \cdots p_k)} \\ &amp; = &amp; 1 + \frac{1}{p_1 - 1} + \dots + \frac{1}{p_k - 1} + \frac{1}{(p_1 - 1)(p_2 - 1)} + \dots + \frac{1}{(p_{k-1}-1)(p_k-1)} \\ &amp; + &amp; \dots + \frac{1}{(p_1 - 1) \cdots (p_k - 1)} \end{eqnarray*}  I am unsure of how to proceed from this point however. Any help would be much appreciated. Changing subject and verb positions in statements and questions My friend and I are planning a vacation to Ireland later this year. We'll be there for a week, flying into Dublin and then renting a car and driving around the country.  Both of us are 23. Are there any age restrictions on renting cars in Ireland? What would be the cheapest/best option for us? How do you return from 'gf' in Vim I am using Vim for windows installed in Unix mode. Thanks to this site I now use the gf command to go to a file under the cursor.  I'm looking for a command to either:   return to the previous file (similar to Ctrl+T for ctags), or   remap gf to automatically launch the new file in a new window. It is asked to solve the PDE  $$u_x + u_y + u = e^{x+2y}$$  My attempt: We have that $$\frac{dx}{1}=\frac{dy}{1}=\frac{du}{e^{x+2y}-u} $$  $\Rightarrow \frac{dy}{dx}=1 \Rightarrow y=x+C_1 \iff C_1=y-x$  $\Rightarrow \frac{du}{dx}= e^{x+2y}-u \iff \dot{u}+u=e^{x+2y} \iff \\ \frac{d}{dx}(e^x u) = e^{2x+2y} \iff u = e^{-x} ( \frac{1}{2}e^{2x+2y} + C_2)$  As $u$ is constant along its characteristics:  $$u(x,y)=\frac{e^{-x}}{2}(e^{2x+2y} + 2C_2(y-x))$$  where $C_2$ is any function of one variable. But wolfram gave the following answer  $$u(x,y)=\frac{e^{-x}}{4}(e^{2x+2y} + 4C_2(y-x))$$  as you can check   What am I doing wrong? Pointer to class data member "::*" How I can create a graph of acceleration and speed - tikz? How can I create a graph of acceleration and speed like the one in the picture. Knowing that they are each other's consequence?    \documentclass{article} \usepackage{tikz}  \begin{document}  \begin{tikzpicture}               \end{tikzpicture} \end{document} Atiyah Macdonald, Exercise 8.3. Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equivalent:   (1) $A$ is Artinian.   (2) $A$ is a finite $k$-algebra.   I have a question in the proof of (1$\Rightarrow$2): By using the structure theorem, we may assume that $(A,m)$ is an Artin local ring. Then $A/m$ is a finite algebraic extension of $k$ by . Since $A$ is Artinian, $m$ is the nilradical of $A$ and thus $m^n=0$ for some $n$. Thus we have a chain $A \supseteq m \supseteq m^2 \supseteq \cdots \supseteq m^n=0$. Since $A$ is Noetherian, $m$ is finitely generated and hence each $m^i/m^{i+1}$ is a finite dimensional $A/m$-vector space, hence a finite dimensional $k$-vector space.  But now how can I deduce that $A$ is a finite dimensional $k$-vector space? How can I take a photo using a slow shutter speed without it being overexposed? When I try to take a picture using a slow shutter speed in daylight it is overexposed. How can I change the settings so it isn't overexposed? Let $D$ be an integral domain and let $a^m=b^m$ and $a^n=b^n$ where $m$ and $n$ are relatively prime integers, $a,b \in D$.   How do I show $a=b$? I had posted a  last week about customizing a ToolValidator class and got some very good answers.  In working with the proposed solutions, I have created a custom module that performs queries on a db, and will be called by both the ToolValidator class (to provide values for the drop-down lists) and later in the geoprocessing script (to get other parameters based on items selected in drop-down lists).  However, I can't seem to actually call the custom module in the ToolValidator class.  I have been trying to append to path with no luck.  When I try to apply these changes to the script, I get a runtime error:  [Errno 9] Bad file descriptor.  If I comment out import line, no errors.  sys.path.append('my_custom_module_directory') import my_custom_module   Many of you may be asking why don't I just implement a custom tool with ArcObjects.  The reason is that my end users don't have the privleges necessary to register ANY dlls on their computer.  UPDATE:  This was happening to me in ArcGIS 10.  Interestingly enough, I was initially appending to the path inside of the initialiazeParameters function of the ToolValidator class.  If I do the append outside (i.e., on top of) of the ToolValidator class, everything works as expected.  sys.path.append('C:/Working/SomeFolder') import somescript --------&gt;THIS WORKS  class ToolValidator:   """Class for validating a tool's parameter values and controlling   the behavior of the tool's dialog."""    def __init__(self):     """Setup arcpy and the list of tool parameters."""     import arcpy     sys.path.append('C:/Working/SomeFolder')     import somescript -------&gt; THIS DOESNT WORK     self.params = arcpy.GetParameterInfo()   UPDATE 2:  I think I found the true cause of my problem.  In the code snippets in this post, I have been appending what appear to be real paths (ex C:/Working/SomeFolder) to the sys.path.  In my actual ToolValidator class, I was constructing a relative path using os.path.dirname(__file__) +"\my_special_folder...".  I was anticipating that os.path.dirname(__file__) would return the path of the toolbox, since that contains the ToolValidator class.  I have come to find this is not the case.  As far as I can tell, the ToolValidator class is never actually written to a .py file, and I speculate the the this code is passed to the python interpreter in memory, so __file__ is useless, or some temp script is persisted and then execfile(path_to_script) is called, again rendering __file__ useless.  I'm sure there are other reasons I am missing as well.    Long story short, if I use a hardcoded path, the sys.append works anywhere, relative paths don't work so well in the ToolValidator class. I want a to be rounded to 13.95.  &gt;&gt;&gt; a 13.949999999999999 &gt;&gt;&gt; round(a, 2) 13.949999999999999   The  function does not work the way I expected. Bug Identification: Tiny Bugs linear transformation $T$ such that $TS = ST$ Let $V$ be a finite-dimensional vector space over $F$. Let $T:V \rightarrow V$ be a linear transformation such that $ST=TS$ for all linear transformations $S:V \rightarrow V$. Show that $T = cI_v$ for some $c \in F$. How to get this blur texture effect in photoshop? Call of Duty MW3 Prestige token lost