In Z40 ⊕ Z30 find two subgroups of order 12

ln⁡ Z40, order(10)=4010=4 ln30, order(10)=3010=3 ln⁡ Z40 ⊕ Z30,order(10, 10)=lcm(4, 3)=12 Thus, ⟨10, 10⟩ is a subgroups of order 12 Similarly, ln⁡ Z40, order(10)=4010=4 ln30, order(5)=305=6 So, in Z40 ⊕ Z30, order(10, 5)=lcm(4, 6)=12 Thus, ⟨10, 5⟩ is a subgroups of order 12

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Statistics and Probability
College Statistics
Comparing two groups

Phoebe

Answered question

2021-01-06

Z_{40} \oplus Z_{30}

find two subgroups of order 12

Answer & Explanation

Aubree Mcintyre

Skilled2021-01-07Added 73 answers

$\ln Z_{40}, o r d e r (10) = \frac{40}{10} = 4$
$\ln_{30}, o r d e r (10) = \frac{30}{10} = 3$
$\ln Z_{40} \oplus Z_{30}, o r d e r (10, 10) = l c m (4, 3) = 12$
Thus, $⟨ 10, 10 ⟩$ is a subgroups of order 12
Similarly,
$\ln Z_{40}, o r d e r (10) = \frac{40}{10} = 4$
$\ln_{30}, o r d e r (5) = \frac{30}{5} = 6$
So, in $Z_{40} \oplus Z_{30}, o r d e r (10, 5) = l c m (4, 6) = 12$
Thus, $⟨ 10, 5 ⟩$ is a subgroups of order 12

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