GATE CS 2024 Set 2 — Question 63
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Engineering Mathematics → Sets & Combinatorics → Groups, Rings & Fields
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Question
Let be the group of integers with addition modulo as the group operation. The number of elements in the group that are their own inverses is ________
Correct answer
4 to 4
Solution
An element in the group is its own inverse if , where is the identity element of the group. This condition is equivalent to solving the following congruences independently for each component:
1.In : . This is satisfied by and . Thus, there are 2 such elements in .
2.In : . Since , the only solution is . Thus, there is 1 such element in .
3.In : . This is satisfied by and . Thus, there are 2 such elements in .
According to the fundamental principle of counting, the total number of elements in the product group that are their own inverses is the product of the number of solutions in each component:The specific elements are .Continue learning with Success Tracker
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