Question

Which of the following is a subgroup of the group $G = \\{1, 2, 3, 4, 5, 6\\}$ under $\otimes_7$ ?

• A. $\\{2, 6, 1\\}$
• B. $\\{1, 2, 4\\}$
• C. $\\{5, 4, 2\\}$
• D. $\\{2, 3 , 1\\}$

Answer 1

Answer: (B)

$\\{1, 2, 4\\}$

Answer 2

$G = \\{1, 2, 3, 4, 5, 6\\}, \oplus_7$
(a) \\{2, 6, 1) is not a subgroup
$\because 2\oplus_7 6=5 \notin \\{ 2,6, 1 \\}$
(b) $\\{1, 2, 4\\}$ is a subgroup
$\because$ it satisfies all properties of group.

$1\oplus_7 2=2, 2 \oplus_7 4=1,4 \oplus_7 1=4$
(c) $\\{5, 4, 2 \\}$ is not a subgroup ?
$\because$ In this set identity element do not exist.
(d) $\\{2, 3, 1\\}$ is not a subgroup
$\because 2 \oplus_7 3 = 6 \notin \\{2, 3, 1\\}$