User falagar - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:45:13Z http://mathoverflow.net/feeds/user/7079 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31821/problems-known-to-be-in-both-np-and-conp-but-not-known-to-be-in-p Problems known to be in both NP and coNP, but not known to be in P falagar 2010-07-14T10:03:22Z 2011-07-14T20:00:26Z <p>One such problem I know is integer factorization.</p> <p>What are other interesting cases?</p> http://mathoverflow.net/questions/31691/inverting-the-totient-function Inverting the totient function falagar 2010-07-13T12:12:57Z 2011-06-03T07:49:49Z <p>For what values of $n$ does the equation $\phi(x) = n$ have at least one solution? Is there any efficient way to check it for a given $n$?</p> <p>It obviously has no solutions for odd $n$. And the smallest even number for which it has no solutions is $14$.</p> http://mathoverflow.net/questions/32126/function-with-range-equal-to-whole-reals-on-every-open-set Function with range equal to whole reals on every open set falagar 2010-07-16T07:05:46Z 2010-12-12T11:55:43Z <p>There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$.</p> <p>I want to generalize this in a way to get a function which is not just unbounded on every open set, but whose range is equal to $\mathbb{R}$ on every open set. The latter construction clearly doesn't work.</p> <p>I'm interested whether such function exists and if it exists is there any constructive way to define it?</p> http://mathoverflow.net/questions/35462/dynamic-programming-and-combinatorics/35478#35478 Answer by falagar for dynamic programming and combinatorics falagar 2010-08-13T11:26:42Z 2010-08-13T11:26:42Z <p>You can use dynamic programming as you suggest in the title.</p> <p>Let $w_{ij}$ be the max gain you can get putting $j$ balls into first $i$ buckets.</p> <p>Then $w_{ij}$ has the following recursive relation: $$w_{i,j} = \max_{0 \le t \le \min(N-1, j)}(w_{i-1,j-t} + \Delta l_{i, t})$$</p> <p>There $t$ is the number of balls you put into $i$-th bucket. Hence you can calculate $w_{K, \lambda}$ (your answer) in time $O(K \lambda N)$.</p> http://mathoverflow.net/questions/35453/finding-all-cycles-of-a-certain-length-in-a-graph/35456#35456 Answer by falagar for Finding all cycles of a certain length in a graph falagar 2010-08-13T09:14:18Z 2010-08-13T09:34:38Z <p>If you consider only simple cycles (every vertex visited at most once) then this problem is NP-complete, so no polynomial (in $|G|$ and $k$) algorithm is known.</p> <p>If non-polynomial algorithms are ok, you can use dynamic programming algorithm with complexity $O(\sum_{i=0}^{i\le k}\binom{n}{i}n^2)$. This algorithm calculates for every subset $S$ of at most $k$ vertices and every vertex $v \in S$ from this subset the number of paths that goes through all vertices from $S$ and has $v$ as the last vertex.</p> http://mathoverflow.net/questions/35339/polynomial-with-prime-powers-values Polynomial with prime powers values falagar 2010-08-12T11:17:06Z 2010-08-12T18:53:44Z <p>It's well known that there are no non-constant polynomials with integer coefficients whose values at integer points are primes. Could this result be generalized to the case of prime powers?</p> <p>The question is whether there exists a polynomial $p(x) \in \mathbb{Z}[x]$ with degree at least one such that for all $x \in \mathbb{Z}$ $|p(x)|$ is prime power.</p> http://mathoverflow.net/questions/35360/consecutive-integers-with-many-prime-factors/35364#35364 Answer by falagar for Consecutive integers with many prime factors falagar 2010-08-12T15:18:30Z 2010-08-12T15:18:30Z <p>Answer for the first question. Let $w_k$ be the product $p_{nk+1} p_{nk+2} \cdots p_{nk+n}$ where $p_i$ is $i$-th prime. So each $w_k$ is equal to product of $n$ distinct primes and $w_k$ and $w_q$ are coprime for $k \ne q$. Now take $t$ such that $t \equiv -k \pmod {w_k}$ for $k = 1, 2, \cdots, m$. Such $t$ exists by Chinese Remainder Theorem. Now $t + k$ has at least $n$ primes because $t + k$ is a multiple of $w_k$. Hence numbers $t + 1, t + 2, \cdots, t + m$ satisfy requirements in the first question.</p> http://mathoverflow.net/questions/33304/mareys-problem-generating-all-prime-numbers-in-n-1-n-2/33307#33307 Answer by falagar for Marey's problem: Generating all prime numbers in $[n_1,n_2]$ falagar 2010-07-25T18:09:05Z 2010-07-25T18:09:05Z <p>If $n_1$ and $n_2$ are not too big you can use Eratosthenes sieve (see wikipedia for this). If they are big enought you can use sieve to cross out numbers which have small prime divisors and after that check every number that wasn't crossed out and check if it is prime using any primality test (see <a href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test" rel="nofollow">http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test</a>).</p> http://mathoverflow.net/questions/32624/special-cases-of-dirichlets-theorem Special cases of Dirichlet's theorem falagar 2010-07-20T12:14:10Z 2010-07-23T22:05:42Z <p>Dirichlet's theorem states that for any coprime $k$ and $m$ there exists infinitely many primes $p$ such that $p \equiv k \pmod m$.</p> <p>Some special cases of this theorem are easy to prove without any analytic methods. Those cases include, for example, $m=4, k=1$ and $m=4, k=3$.</p> <p>Both cases could be proved by considering first $t$ prime numbers $p_i \equiv k \pmod m$ and constructing a new number which is proved to have prime divisor $p \equiv k \pmod m$ that is not equal to any $p_i$.</p> <p>For case $m=4, k=1$ we can consider number $(p_1 p_2 \cdots p_t)^2 + 1$. And for case $m=4, k=3$ number $4p_1 p_2 \cdots p_k + 3$.</p> <p>Those constructions could also be applied to some other special cases as well.</p> <p>Are there any other special cases for which there exists a simple non-analytic proof which don't use any of those two constructions?</p> http://mathoverflow.net/questions/32771/does-there-exist-a-pure-recurrence-formula-with-polynomial-coefficients-for-fibon/32776#32776 Answer by falagar for Does there exist a pure recurrence formula with polynomial coefficients for Fibonacci(2^n)? falagar 2010-07-21T10:10:02Z 2010-07-23T08:21:09Z <p>There is the following formula: $$x_{n+2} = \frac{x_{n+1}^3}{2 x_{n}^2} + \frac{5}{2} x_n^2 x_{n+1}$$ I'm not sure if this is a pure recurrence formulae. If you need, I may provide a proof.</p> http://mathoverflow.net/questions/32956/representation-of-rational-numbers-as-the-sum-of-1-k Representation of rational numbers as the sum of 1/k falagar 2010-07-22T13:54:47Z 2010-07-22T13:54:47Z <p>Is it true that every positive rational number $r = \frac{n}{m}$ could be represented as sum of $\frac{1}{k}$ for different $k$'s:</p> <p>$$r = \sum_{i=1}^{s} \frac{1}{c_i}$$ where all $c_i$ are different?</p> <p>If true, are there any bounds on number of summands $s$ based on $n$ and $m$?</p> http://mathoverflow.net/questions/32801/is-there-a-two-party-multiplicative-and-additive-secret-sharing-scheme/32804#32804 Answer by falagar for Is there a two-party multiplicative and additive secret sharing scheme ? falagar 2010-07-21T14:32:17Z 2010-07-21T14:32:17Z <p>Yes, there is such scheme. It was recently suggested by Craig Gentry.</p> <p>Reference: "Fully Homomorphic Encryption Using Ideal Lattices" by Craig Gentry. <a href="http://domino.research.ibm.com/comm/research_projects.nsf/pages/security.homoenc.html/" rel="nofollow">http://domino.research.ibm.com/comm/research_projects.nsf/pages/security.homoenc.html/</a>$FILE/stocdhe.pdf</p> http://mathoverflow.net/questions/32761/number-of-subset-sums/32768#32768 Answer by falagar for Number of subset sums falagar 2010-07-21T09:10:09Z 2010-07-21T09:10:09Z <p>There are$\binom{n}{k}$ways to choose$k$elements from$n$elements. If we consider their sum it is "expected" to be equal to every element of the field with the same probability. Hence we get$\frac{1}{q}\binom{n}{k}$for the number of solutions of$x_1 + x_2 + \cdots + x_k = s$. This is the first part of your question.</p> <p>Now there are$\binom{q}{k}$ways to choose$k$different elements of the field. And$\frac{1}{q}$of them have sum equal to$s$. In other words number of solutions of$x_1 + x_2 + \cdots + x_k = s$is equal to$\frac{1}{q}\binom{q}{k}$.</p> <p>If we choose random$D$with$|D|=n$every solution will have all$x_i \in D$with the same probability$p$:</p> <p>$$p = \frac{\binom{q-k}{n-k}}{\binom{q}{n}}$$</p> <p>Hence the expected number of solutions is equal to$\frac{1}{q}\binom{q}{k} \cdot p$which is equal to$\frac{1}{q}\binom{n}{k}$.</p> <p>So the expected number of solutions is equal to$\frac{1}{q}\binom{n}{k}$.</p> http://mathoverflow.net/questions/32502/cesaro-convergence-implies-weak-convergence-of-a-subsequence/32508#32508 Answer by falagar for Cesaro convergence implies weak convergence of a subsequence falagar 2010-07-19T14:58:22Z 2010-07-19T14:58:22Z <p>If we take$x_n = (-1)^n x$then$x_n$converges to$0$in Cesaro sence. But no subsequence of$x_n$converges weakly to$0$.$x_n$is also a bounded sequence. Hence your statements seems wrong.</p> http://mathoverflow.net/questions/32269/guess-a-number-with-at-most-one-wrong-answer/32270#32270 Answer by falagar for Guess a number with at most one wrong answer falagar 2010-07-17T08:00:47Z 2010-07-17T08:00:47Z <p>Yes, there is a way to guess a number asking <strong>14</strong> questions in worst case. To do it you need a linear code with length 14, dimension 10 and distance at least 3. One such code can be built based on Hamming code (see <a href="http://en.wikipedia.org/wiki/Hamming_code" rel="nofollow">http://en.wikipedia.org/wiki/Hamming_code</a>).</p> <p>Here is the strategy.</p> <p>Let us denote bits of first player's number as$a_i$,$i \in [1..10]$. We start with asking values of all those bits. That is we ask the following questions: "is it true that i-th bit of your number is zero?" Let us denote answers on those questions as$b_i$,$i \in [1..10]$.</p> <p>Now we ask <strong>4</strong> additional questions:</p> <p>Is it true that$a_{1} \otimes a_{2} \otimes a_{4} \otimes a_{5} \otimes a_{7} \otimes a_{9}$is equal to zero? ($\otimes$is sumation modulo$2$).</p> <p>Is it true that$a_{1} \otimes a_{3} \otimes a_{4} \otimes a_{6} \otimes a_{7} \otimes a_{10}$is equal to zero?</p> <p>Is it true that$a_{2} \otimes a_{3} \otimes a_{4} \otimes a_{8} \otimes a_{9} \otimes a_{10}$is equal to zero?</p> <p>Is it true that$a_{5} \otimes a_{6} \otimes a_{7} \otimes a_{8} \otimes a_{9} \otimes a_{10}$is equal to zero?</p> <p>Let$q_1$,$q_2$,$q_3$and$q_4$be answers on those additional questions. Now second player calculates$t_{i}$($i \in [1..4]$) --- answers on those questions based on bits$b_j$which he previously got from first player. </p> <p>Now there are 16 ways how bits$q_i$can differ from$t_i$. Let$d_i = q_i \otimes t_i$(hence$d_i = 1$iff$q_i \ne t_i$).</p> <p>Let us make table of all possible errors and corresponding values of$d_i$:<br /> position of error ->$(d_1, d_2, d_3, d_4)$<br /></p> <p>no error -> (0, 0, 0, 0)<br /> error in$b_1$-> (1, 1, 0, 0)<br /> error in$b_2$-> (1, 0, 1, 0)<br /> error in$b_3$-> (0, 1, 1, 0)<br /> error in$b_4$-> (1, 1, 1, 0)<br /> error in$b_5$-> (1, 0, 0, 1)<br /> error in$b_6$-> (0, 1, 0, 1)<br /> error in$b_7$-> (1, 1, 0, 1)<br /> error in$b_8$-> (0, 0, 1, 1)<br /> error in$b_9$-> (1, 0, 1, 1)<br /> error in$b_{10}$-> (0, 1, 1, 1)<br /> error in$q_1$-> (1, 0, 0, 0)<br /> error in$q_2$-> (0, 1, 0, 0)<br /> error in$q_3$-> (0, 0, 1, 0)<br /> error in$q_4$-> (0, 0, 0, 1)<br /></p> <p>All the values of$(d_1, d_2, d_3, d_4)$are different. Hence we can find where were an error and hence find all$a_i$.</p> http://mathoverflow.net/questions/31797/finite-groups-with-elements-of-order-n Finite groups with elements of order n falagar 2010-07-14T05:24:20Z 2010-07-14T11:20:15Z <p>Consider a finite group where all elements have the same order$n$. What could be said about such groups?</p> <p>For$n=2$it could be proved that such group is isomorphic to$(\mathbb{Z}/2\mathbb{Z})^k$. Could it be somehow generalized on case$n>2$?</p> <p><strong>EDIT:</strong> Surely the identity has order 1, so we have to exclude it.</p> http://mathoverflow.net/questions/31814/checking-whether-given-binary-operation-is-a-group-operation Checking whether given binary operation is a group operation falagar 2010-07-14T09:08:31Z 2010-07-14T09:31:56Z <p>Given a binary function$f: [1..n] \times [1..n] \to [1..n]$how to check that this operation is a group operation on$[1..n]$?</p> <p>It's obvious that this can be done in$O(n^3)$time just by checking all group properties. The most time-expensive property is associativity. Also it's clear that it could not be done faster than$O(n^2)$time since you should at least examine all values$f(i,j)$.</p> <p>The question is if there is any algorithm to solve this problem in time faster than$O(n^3)$?</p> http://mathoverflow.net/questions/31553/finite-nonabelian-groups-of-odd-order Finite nonabelian groups of odd order falagar 2010-07-12T14:06:10Z 2010-07-14T04:41:09Z <p>For every even$n$there exists nonabelian group. As example of such group we can take dihedral group.</p> <p>The question is about odd$n$. For some of them there are no nonabelian groups of order$n$(for example, if$n$is prime then the group of order$n$is cyclic and hence abelian).</p> <p>For what odd$n$are there known examples of nonabelian finite groups of order$n$?</p> http://mathoverflow.net/questions/31669/interchanging-summations/31676#31676 Answer by falagar for Interchanging summations falagar 2010-07-13T08:49:46Z 2010-07-13T08:49:46Z <p>All the terms in this summation are positive, so you can change the order. The result wouldn't change.</p> <p>The summation is done over all pairs$(k, j)$with$j \le k$. The first summation first sums over all$k$and after that over all$j$, the second one --- first sums over all$j$and after that over all$k$.</p> http://mathoverflow.net/questions/31663/distinct-numbers-in-multiplication-table Distinct numbers in multiplication table falagar 2010-07-13T05:37:54Z 2010-07-13T06:06:21Z <p>Consider multiplication table for numbers$1,2,\cdots, n$. How many different numbers are there? That is how many different numbers of the form$ij$with$1 \le i, j \le n$are there?</p> <p>I'm interested in a formulae or an algorithm to calculate this number in time less than$O(n^2)$.</p> http://mathoverflow.net/questions/15614/interesting-applications-of-max-flow-and-linear-programming/31556#31556 Answer by falagar for Interesting applications of max-flow and linear programming falagar 2010-07-12T14:23:22Z 2010-07-12T14:23:22Z <p>Another interesting application of LP is finding Nash equilibrium for a two player zero-sum game.</p> <p><a href="http://en.wikipedia.org/wiki/Zero-sum_game#Solving" rel="nofollow">http://en.wikipedia.org/wiki/Zero-sum_game#Solving</a></p> http://mathoverflow.net/questions/31271/approximation-with-continuous-functions Approximation with continuous functions falagar 2010-07-10T05:58:43Z 2010-07-12T13:35:40Z <p>Is it true that for every function$\mathbb{R} \to \mathbb{R}$there exists a sequence of continuous functions$f_n(x): \mathbb{R} \to \mathbb{R}$such that for any$x \in \mathbb{R}f_n(x)$converges to$f(x)$?</p> <p>I started with characteristic function of rationals and tried to find corresponding sequence and got stuck. So additional question if this statement is true for this function.</p> http://mathoverflow.net/questions/31505/math-and-music-theory Math and music theory falagar 2010-07-12T06:20:33Z 2010-07-12T06:20:33Z <p>Are there any interesting connections between math objects (theorems, definitions, constructions) and music theory?</p> http://mathoverflow.net/questions/26079/algorithm-for-determining-if-a-path-exists-in-a-graph-or-if-not-the-closest-edit/31296#31296 Answer by falagar for Algorithm for determining if a path exists in a graph or if not, the closest edit distance. falagar 2010-07-10T13:40:46Z 2010-07-10T13:40:46Z <p>If graph is acyclic you can use some sort of dynamic programming. Let$a_{u,k}$be the best distance you can get if you start from vertex$u \in G$and consider only$k$last vertices of your given path.</p> <p>It's quite straightforward how to calculate all$a_{u,k}$based on all values of$a_{u',k'}$with$u'$"after"$u$and$k' &lt; k$: you just iterate over all edges going from vertex$u$.</p> <p>This approach works in$O(|G| \times strlen)$time.</p> http://mathoverflow.net/questions/30874/arithmetic-fixed-point-theorem/31291#31291 Answer by falagar for Arithmetic fixed point theorem falagar 2010-07-10T12:54:50Z 2010-07-10T12:54:50Z <p>Another approach, I suppose, is to consider this as a kind of <strong>Cantor's diagonal argument</strong>. If you write a table with rows correspond to formula and columns correspond to integers then$sub([F],[t])=[F_v(t)]$is just a formulae in$[F]$row and$t$column. Now, following Cantor's diagonal argument, you might want to consider diagonal of this table, which is$sub(v,num(v))$and augment it somehow. Here is there$F$comes in - you apply it to the diagonal and get$H(v)=F_v(sub(v,num(v)))$. Why should you? I don't know exactly. But it's similar to standard Cantor's argument.</p> <p>Hence,$H(v)$is a formulae with the following property: for every formulae$T(v)$there exist$n$such that$H(n) \Leftrightarrow F(T(n))$(because of diagonal argument --- take row number$[T]$and intersect it with diagonal). That's why we can take$n = [T]$.</p> <p>Hence, if we take$T = H$, and$n = [T] = [H]$we would get what we need.</p> http://mathoverflow.net/questions/31163/unbounded-linear-operator-defined-on-l2 Unbounded linear operator defined on$l^2$falagar 2010-07-09T09:08:35Z 2010-07-09T12:27:08Z <p>Let$l^2$be a Hilbert space of infinite sequences$(z_0, z_1, \cdots)$with finite$\sum_{i=0}^{\infty} |z_i|^2$.</p> <p>Are there any simple example of unbounded linear opearator$T: l^2 \to l^2$with$D(T)=l^2$?</p> http://mathoverflow.net/questions/31143/unique-integer-partitions/31164#31164 Answer by falagar for unique integer partitions falagar 2010-07-09T09:29:39Z 2010-07-09T09:29:39Z <p>If you need an algorithm to calculate this number, you can use the following. Let$a_{nk}$be an answer to your question, then it's not hard to prove that$a_{nk} = a_{n,k-1} + a_{n - k, k}$. So you can fill in the table of all$a_{nk}$using this formulae.</p> http://mathoverflow.net/questions/35464/partition-of-a-number Comment by falagar falagar 2010-08-13T11:54:01Z 2010-08-13T11:54:01Z @pravesh, in your example$de &lt; p$. http://mathoverflow.net/questions/35462/dynamic-programming-and-combinatorics/35478#35478 Comment by falagar falagar 2010-08-13T11:52:15Z 2010-08-13T11:52:15Z And I don't know if there exists faster solution. http://mathoverflow.net/questions/35462/dynamic-programming-and-combinatorics/35478#35478 Comment by falagar falagar 2010-08-13T11:50:31Z 2010-08-13T11:50:31Z Is$R(p)$the optimal gain in putting$p$balls considering all the buckets? If so then I don't understand your formula. If you have put$i$balls into$k$-th bucket you can't use this bucket any more. http://mathoverflow.net/questions/35464/partition-of-a-number Comment by falagar falagar 2010-08-13T10:01:23Z 2010-08-13T10:01:23Z What if$l=m=1$and$N &gt; p+1$? Then you take partitions$p=p$and$p+1=p+1$and there are no subsets with equal sums. http://mathoverflow.net/questions/35453/finding-all-cycles-of-a-certain-length-in-a-graph/35456#35456 Comment by falagar falagar 2010-08-13T09:50:56Z 2010-08-13T09:50:56Z Yes, if$k$is fixed and we are interested only how time complexity depends on$n$then both algorithms have the same complexity. They actually differ by a factor of$k!$. http://mathoverflow.net/questions/35453/finding-all-cycles-of-a-certain-length-in-a-graph/35456#35456 Comment by falagar falagar 2010-08-13T09:36:27Z 2010-08-13T09:36:27Z The number of$k$-tuples with distinct elements is$\frac{n!}{(n-k)!}$which for big$k$is much more than the number of subsets with at most$k$elements. For example if$n=k$then this algorithm works in time$O(n^2 2^n)$and enumerating algorithm in time$O(n!)$. http://mathoverflow.net/questions/33304/mareys-problem-generating-all-prime-numbers-in-n-1-n-2 Comment by falagar falagar 2010-07-26T09:46:47Z 2010-07-26T09:46:47Z There are too many prime numbers between$n_1$and$n_2$in those cases. Even for the first case there are about$10^12$primes. http://mathoverflow.net/questions/33088/entropy-of-a-general-prob-measure Comment by falagar falagar 2010-07-23T13:10:36Z 2010-07-23T13:10:36Z I can suggest <a href="http://en.wikipedia.org/wiki/Differential_entropy" rel="nofollow">en.wikipedia.org/wiki/Differential_entropy</a> http://mathoverflow.net/questions/32771/does-there-exist-a-pure-recurrence-formula-with-polynomial-coefficients-for-fibon/32776#32776 Comment by falagar falagar 2010-07-23T08:21:50Z 2010-07-23T08:21:50Z tdnoe, thank you for your comment! It indeed was a typo. http://mathoverflow.net/questions/32956/representation-of-rational-numbers-as-the-sum-of-1-k Comment by falagar falagar 2010-07-23T08:18:11Z 2010-07-23T08:18:11Z Thank you for your comments! I think we should close the question. http://mathoverflow.net/questions/32761/number-of-subset-sums/32768#32768 Comment by falagar falagar 2010-07-21T10:45:26Z 2010-07-21T10:45:26Z I think that I don't use the assumption that q is comprime to k. http://mathoverflow.net/questions/32624/special-cases-of-dirichlets-theorem Comment by falagar falagar 2010-07-20T13:06:25Z 2010-07-20T13:06:25Z Actually I want proof that is niether of two mentioned in the question. By &quot;a simple non-analytic proof&quot; I mean a proof that doesn't use any of classical analysis. http://mathoverflow.net/questions/31821/problems-known-to-be-in-both-np-and-conp-but-not-known-to-be-in-p Comment by falagar falagar 2010-07-14T11:30:18Z 2010-07-14T11:30:18Z Integer factorization can be converted into decision problem: given$n$and$k$return YES if$k$-th bit of the smallest prime divisor of$n$is equal to$1$. http://mathoverflow.net/questions/31796/exptime-algorithms Comment by falagar falagar 2010-07-14T06:10:12Z 2010-07-14T06:10:12Z Do you need just EXPTime-Complete problem? If so there is one in wikipedia: <a href="http://en.wikipedia.org/wiki/EXPTIME" rel="nofollow">en.wikipedia.org/wiki/EXPTIME</a>. Given a (determinister for first question, nondeterministic for the second) Turing machine and time T decide wheter it halts after T steps. http://mathoverflow.net/questions/31797/finite-groups-with-elements-of-order-n Comment by falagar falagar 2010-07-14T05:59:50Z 2010-07-14T05:59:50Z Well, I understand now that$n$must be prime and group might be nonabelian. So there are no straitforward generalizations of case$n=2\$. Originally, I was looking for the statement like '... then the group is isomorphic to such or such group'. Now I think there are no such classification and the question is closed.