GATE CS 2014 Set 3 — Question 60
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Question
There are two elements in a group such that every element in the group can be written as a product of some number of 's and 's in some order. It is known thatwhere is the identity element. The maximum number of elements in such a group is __________.
Correct answer
4 to 4
Solution
Given the group is generated by and , so every element is a product of 's and 's.
The relations are given as:
(since ).
Multiplying by on the right:
(since ).Since , the generators commute, meaning the group is abelian. Any element in the group can be written in the form .
Using the relations and , the exponents and can only be or (modulo 2).The possible distinct elements are:
The relations are given as:
1.
2.
3.
4.
From , we have . Multiplying by on the right:(since ).
Multiplying by on the right:
(since ).Since , the generators commute, meaning the group is abelian. Any element in the group can be written in the form .
Using the relations and , the exponents and can only be or (modulo 2).The possible distinct elements are:
1.
2.
3.
4.
Thus, the maximum number of elements in such a group is 4. This group is isomorphic to the Klein four-group ().Continue learning with Success Tracker
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