The set {1,2,4,7,8,11,13,14} forms a group under the operationof multiplication

Question

The set {1,2,4,7,8,11,13,14} forms a group under the operationof multiplication modulo 15. Which of the following is the cyclicsubgroup generated by {7}. a.) {1,7} b.) {1,2,7,14} c.) {1,4,7,13} d.) {1,7,8,13,14} e.) {4,7,11,14} Please explain answer. I'm weak in this area. Thanks The set {1,2,4,7,8,11,13,14} forms a group under the operationof multiplication modulo 15. Which of the following is the cyclicsubgroup generated by {7}. a.) {1,7} b.) {1,2,7,14} c.) {1,4,7,13} d.) {1,7,8,13,14} e.) {4,7,11,14} Please explain answer. I'm weak in this area. Thanks

Explanation / Answer

The cyclic group is generated by computation ... so compute [7^1] = [7]... and continue with [7^n] ... until getting back to[7]. [7^2] = [49] = [ 45 + 4 ] = [ 4]. [7^3] = 343 = [330 + 13] = [13]. Note that 15 * 22 =330. [7^4] = 2401 = [2400 + 1 ] = [1]. Note that 2400 / 3 = 800,and 800 is a multiple of 5, so 2400 is a multiple of 15. [7^5] = [16807] = [16800 + 7] = [7]. Note again that 16800/3 = 5600which is multiple of 5. So the cyclic group is { 7, 4, 13, 1} which is answer c.

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