CCC '06 S4 - Groups - Olympiads Online Judge

CCC '06 S4 - Groups

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Points: 10 (partial)

Time limit: 1.0s

Memory limit: 256M

Problem type

Canadian Computing Competition: 2006 Stage 1, Senior #4

In mathematics, a group, $G$ $G$ , is an object that consists of a set of elements and an operator (which we will call $\times$ $\times$ ) so that if $x$ $x$ and $y$ $y$ are in $G$ $G$ so is $x \times y$ $x \times y$ . Operations also have the following properties:

Associativity: For all $x$ $x$ , $y$ $y$ and $z$ $z$ in $G$ $G$ , $x \times (y \times z) = (x \times y) \times z$ $x \times (y \times z) = (x \times y) \times z$ .
Identity: the group contains an "identity element" (we can use $i$ $i$ ) so that for each $x$ $x$ in $G$ $G$ , $x \times i = x$ $x \times i = x$ and $i \times x = x$ $i \times x = x$ .
Inverse: for every element $x$ $x$ there is an inverse element (we denote by $x^{-1}$ $x^{- 1}$ ) so that $x \times x^{-1} = i$ $x \times x^{- 1} = i$ and $x^{-1} \times x = i$ $x^{- 1} \times x = i$ .

Groups have a wide variety of applications including the modeling of quantum states of an atom and the moves in solving a Rubik's cube puzzle. Clearly, the integers under addition from a group ( $0$ $0$ is the identity, and the inverse of $x$ $x$ is $-x$ $- x$ , and you can prove associativity as an exercise), though that group is infinite and this problem will deal only with finite groups.

One simple example of a finite group is the integers modulo $10$ $10$ under the operation addition.

That is, the group consists of the integers $0, 1, \dots, 9$ $0, 1, \dots, 9$ and the operation is to add two keeping only the least significant digit. Here the identity is $0$ $0$ . This particular group has the property that $x \times y = y \times x$ $x \times y = y \times x$ , but this is not always the case. Consider the group that consists of the elements $a$ $a$ , $b$ $b$ , $c$ $c$ , $d$ $d$ , $e$ $e$ and $i$ $i$ . The "multiplication table" below defines the operations. Note that each of the required properties is satisfied (associativity, identity and inverse) but, for example, $c \times d = a$ $c \times d = a$ while $d \times c = b$ $d \times c = b$ .

$\displaystyle \begin{array}{c|cccccc} \times & i & a & b & c & d & e \\\hline i & i & a & b & c & d & e \\ a & a & i & d & e & b & c \\ b & b & e & i & d & c & a \\ c & c & d & e & i & a & b \\ d & d & c & a & b & e & i \\ e & e & b & c & a & i & d \end{array}$ $\begin{array}{ccccccc} \times & i & a & b & c & d & e \\ i & i & a & b & c & d & e \\ a & a & i & d & e & b & c \\ b & b & e & i & d & c & a \\ c & c & d & e & i & a & b \\ d & d & c & a & b & e & i \\ e & e & b & c & a & i & d \end{array}$

Your task is to write a program which will read a sequence of multiplication tables and determine whether each structure defined is a group.

Input Specification

The input will consist of a number of test cases. Each test case begins with an integer $n$ $n$ $(0 \le n \le 100)$ $(0 \leq n \leq 100)$ . If the test case begins with $n = 0$ $n = 0$ , the program terminates. To simplify the input, we will use the integers $1, \dots, n$ $1, \dots, n$ to represent elements of the candidate group structure; the identity could be any of these (i.e., it is not necessarily the element $1$ $1$ ). Following the number $n$ $n$ in each test case are $n$ $n$ lines of input, each containing integers in the range $[1, \dots, n]$ $[1, \dots, n]$ . The $q^{th}$ $q^{t h}$ integer on the $p^{th}$ $p^{t h}$ line of this sequence is the value $p \times q$ $p \times q$ .

Output Specification

If the object is a group, output yes (on its own line), otherwise output no (on its own line). You should not output anything for the test case where $n = 0$ $n = 0$ .

Sample Input

Copy

2
1 2
2 1
6
1 2 3 4 5 6
2 1 5 6 3 4
3 6 1 5 4 2
4 5 6 1 2 3
5 4 2 3 6 1
6 3 4 2 1 5
7
1 2 3 4 5 6 7
2 1 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
5 1 1 1 1 1 1
6 1 1 1 1 1 1
7 1 1 1 1 1 1
3
1 2 3
3 1 2
3 1 2
0

Sample Output

Copy

yes
yes
no
no

Explanation

The first two collections of elements are in fact groups (that is, all properties are satisfied). For the third candidate, it is not a group, since $3 \times (2 \times 2) = 3 \times 1 = 3$ $3 \times (2 \times 2) = 3 \times 1 = 3$ but $(3 \times 2) \times 2 = 1 \times 2 = 2$ $(3 \times 2) \times 2 = 1 \times 2 = 2$ . In the last candidate, there is no identity, since $1$ $1$ is not the identity, since $2 \times 1 = 3$ $2 \times 1 = 3$ (not $2$ $2$ ), and $2$ $2$ is not the identity, since $2 \times 1 = 3$ $2 \times 1 = 3$ (not $1$ $1$ ) and $3$ $3$ is not the identity, since $1 \times 3 = 3$ $1 \times 3 = 3$ (not $1$ $1$ ).

Comments

-5

georgezhang006 commented on July 28, 2020, 4:47 p.m.

Are the Operators +, -, *, / only or does it include %(mod)?

This comment is hidden due to too much negative feedback. Show it anyway.

1

littlecat commented on Aug. 11, 2020, 5:00 p.m. ← edited →

The operators are simply 2 variable functions, not necessarily arithmetic functions. For example, in the 6 element group (iabcde), there is no mapping of the elements to (1,2,3,4,5,6) with an arithmetic function. (I think it is isomorphic to the group of symmetries of a triangle as well as the group of permutations with 3 elements.)

3

sushi commented on June 29, 2020, 9:44 p.m.

For those that are a bit confused about this problem, I have a few tips:

The $i$ $i$ described in the inverse definition is actually the identity element described in the identity definition, they are the same variable.

The grid is a times table, so a hint would be that $(x \times y)$ $(x \times y)$ is equivalent as the element at position $(x, y)$ $(x, y)$ in the grid.

An identity element could be $0$ $0$ as hinted in the problem statement, use indexes as the elements to cover that case.

This problem is surprisingly annoying, and I hope this helps.

1

littlecat commented on Aug. 11, 2020, 6:18 p.m. ← edited →

In the input specification, the elements are given as $1$ $1$ to $n$ $n$ , so the elements can be used directly without indexes. I think there is no way to solve the problem without testing all triples $(a, b, c)$ $(a, b, c)$ for the associativity, so that the solution would take $\mathcal{O}(n^3)$ $O (n^{3})$ time. In C++, a vector<vector<int>> also works for recording the function definition.