- #1
Oxymoron
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Question 1
Determine the quotient group [tex](\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle[/tex]
Answer
[tex]\langle(1,2)\rangle[/tex] is a cyclic subgroup [tex]H[/tex] of [tex]\mathbb{Z}_2\times\mathbb{Z}_4[/tex] generated by [tex](1,2)[/tex]. Thus
[tex]H=\{(0,0),(1,2)\}[/tex]
Since [tex]\mathbb{Z}_2\times\mathbb{Z}_4[/tex] has 2.4 = 8 elements, and [tex]H[/tex] has 2 elements, all cosets of [tex]H[/tex] must have 2 elements, and [tex](\mathbb{Z}_2\times\mathbb{Z}_4)/H[/tex] must have order 4.
Possible abelian groups of order 4 are
[tex]\mathbb{Z}_2\times\mathbb{Z}_2[/tex]
[tex]\mathbb{Z}\times\mathbb{Z}_4[/tex]
But I don't know how to work out which one is isomorphic to [tex](\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle[/tex]
Determine the quotient group [tex](\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle[/tex]
Answer
[tex]\langle(1,2)\rangle[/tex] is a cyclic subgroup [tex]H[/tex] of [tex]\mathbb{Z}_2\times\mathbb{Z}_4[/tex] generated by [tex](1,2)[/tex]. Thus
[tex]H=\{(0,0),(1,2)\}[/tex]
Since [tex]\mathbb{Z}_2\times\mathbb{Z}_4[/tex] has 2.4 = 8 elements, and [tex]H[/tex] has 2 elements, all cosets of [tex]H[/tex] must have 2 elements, and [tex](\mathbb{Z}_2\times\mathbb{Z}_4)/H[/tex] must have order 4.
Possible abelian groups of order 4 are
[tex]\mathbb{Z}_2\times\mathbb{Z}_2[/tex]
[tex]\mathbb{Z}\times\mathbb{Z}_4[/tex]
But I don't know how to work out which one is isomorphic to [tex](\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle[/tex]