Consider the alternating group A4 = {σ ε S4 :σ is an even permutation).Which of the following is FALSE?a)A4 has 12 elementsb)A4 has exactly on subgroup of order 4c)A4 has a subgroup of order 6d)Number of 3 cycles in A4 is 8Correct answer is option 'C'. Can you explain this answer?

Question

Consider the alternating group A4 = {&sigma; &epsilon; S4 :&sigma; is an even permutation).Which of the following is FALSE?a)A4 has 12 elementsb)A4 has exactly on subgroup of order 4c)A4 has a subgroup of order 6d)Number of 3 cycles in A4 is 8Correct answer is option 'C'. Can you explain this answer?

Answer

(), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) }.

The first element is the identity element, denoted as (). This element does not change any elements in the group when it is applied, so it is called the identity element.

The second element is a 2-cycle, denoted as (1 2)(3 4). This element swaps the elements 1 and 2, and also swaps the elements 3 and 4. It does not change the elements 5 and 6, so it is called a 2-cycle.

The third element is also a 2-cycle, denoted as (1 3)(2 4). This element swaps the elements 1 and 3, and also swaps the elements 2 and 4. It does not change the elements 5 and 6.

The fourth element is also a 2-cycle, denoted as (1 4)(2 3). This element swaps the elements 1 and 4, and also swaps the elements 2 and 3. It does not change the elements 5 and 6.

These four elements generate the alternating group A4, which consists of all possible permutations of 4 elements that can be obtained by composing these elements.

Answer

Given A_{4 =}

is an even permutation}
A₄ has a subgroup of order 6 is false.
Because in this example converse of Lagrange’s theorem fails.
Hence, option (c) is correct.

Similar Mathematics Doubts

Consider the alternative groupisan even permutation}. Which ofthe following is TRUE ?

Consider the following statements:1. A group of order 289 is abelian.2. In S3 there are four elements satisfying x2= e and six elements satisfying y3 = e.3. Every proper subgroup of S3 is cyclic.4. (Z30, has 4 subgroups of order 15 .Correct statements are

G is a group of order 51. Then which one of the following statements is false?

Let S denote the group of all permutations on the finite set {1, 2, 3, 4, 5} under the operation of permutation multiplication. Then the order of the subgroup of S5 generated byis

Let G = {nε Z : 1 ≤ n ≤ 55, gcd(n, 56) = 1} be a multiplicative group modulo 56. Consider the sets S1- {1,9, 17,25,33,41} and S2 = { 1 ,1 5 ,2 9 ,43 }Q.which one of the following is TRUE?

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams