Question

How many subgroups of order 4 does D_4 have?

Abstract algebra
ANSWERED
asked 2020-12-15
How many subgroups of order 4 does \(\displaystyle{D}_{{4}}\) have?

Answers (1)

2020-12-16
We know that \(\displaystyle{D}_{{n}}\) is a dehedral group of order 2n. It is non-abelian group.
Therefore, the order of \(\displaystyle{D}_{{4}}\) is 8 and the list of elements is given by
\(\displaystyle{D}_{{4}}={\left\lbrace{e},{a},{a}^{{2}},{a}^{{3}},{b},{b}{a},{b}{a}^{{2}},{b}{a}^{{3}}\right\rbrace}\)
There are 3 subgroup of order in \(\displaystyle{D}_{{4}}\) which is given by
\(\displaystyle{H}_{{1}}={\left\lbrace{e},{a},{a}^{{2}},{a}^{{3}}\right\rbrace},\)
\(\displaystyle{H}_{{2}}={\left\lbrace{e},{b},{a}^{{2}},{b}{a}^{{2}}\right\rbrace}\)
\(\displaystyle{H}_{{3}}={\left\lbrace{e},{b}{a},{a}^{{2}},{b}{a}^{{3}}\right\rbrace}\)
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