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)

$G = \{1, 2, 3, 4, 5, 6\}, \oplus_7$
(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
$\because 2 \oplus_7 3 = 6 \notin \{2, 3, 1\}$