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Without using the property of finite abelian group how to evaluate the number of subgroups of order $4$ of $\mathbb Z_4\oplus\mathbb Z_2?$

Please help ! I can show that $\mathbb Z_4\oplus\mathbb Z_2$ has xactly $4$ elements of order $4.$ But that didn't help much since $\mathbb Z_4\oplus\mathbb Z_2$ is non-cyclic.

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Hint: Once you have a firm understanding of the elements of order $4$, you are nearly finished. List the distinct subgroups generated by these. Any other subgroup of order $4$ must consist of elements of order $2$ or $1$. And $\mathbb{Z}_4$ has very few elements of order $2$.

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  • $\begingroup$ An small program done in GAP's environment shows the result: z:=CyclicGroup(4);; n:=CyclicGroup(2);; s:=DirectProduct( z, n );; e:=Subgroups(s);Size(e); Filtered(e,t->Order(t)=4);; 8 $\endgroup$ – mrs Apr 25 '13 at 6:33
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The observation that $\mathbb Z_4\oplus\mathbb Z_2$ has $4$ elements of order $4$ at least helps you to calculate the number of the cyclic subgroups of order $4.$

  • Cyclic subgroups of order $4:$

    Any cyclic subgroup of $\mathbb Z_4\oplus\mathbb Z_2$ of order $4$ has $\phi(4)=2$ generators. So a cyclic subgroup of order $4$ absorbs the elements of order $4$ in pair none of which belong to any other cyclic subgroup of same order.$^\diamond$

    Therefore $\mathbb Z_4\oplus\mathbb Z_2$ has $2$ cyclic group of order $4.$

$\diamond$ For a cyclic group $G=(a)$ of order $n,$ $a^k$ is a generator of $G\iff(n,k)=1.$ Consequently the list of all generators of $G=\{e,a,a^2,...,a^{n-1}\}$ is $\{a^k:(k,n)=1\}$ whence a finite cyclic group of order $n$ has $\phi(n)$ generators.

  • Non-cyclic subgroups of order $4:$

    Of course in a non-cyclic subgroups of order $4$ each non-identity element is of order $2.$ It's an easy exercise to see that there're exactly $3$ such elements.

    Consequently $\mathbb Z_4\oplus\mathbb Z_2$ has $1$ non-cyclic group of order $4.$

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1:Any element of order $4$ generates a cyclic subgroup of order $4$.

2:The three elements of order $2$ including $(0, 0)$ also form a subgroup which is isomorphic to the Klein four group. Your task is to find out the elements of $2 $ and $4$ only.

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0
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Hint: Write the elements in two coordinates and they can be multiplied coordinatewise.

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