Supose that G is a finite abelian group that does not contain a subgro

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  • #1
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let us assume G is not cyclic. Let a be an element of G of maximal order. Since G is not cyclic we have <a>≠G. Let b be an element in G, but not in the cyclic subgroup generated by a.

O(a) = m and O(b) = n where O() refers tothe orders. . then how can we use this to construct a subgroup of G isomorphic to Zp×Zp?

Help will be appreciated. Thank you
 

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  • #2
Dick
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let us assume G is not cyclic. Let a be an element of G of maximal order. Since G is not cyclic we have <a>≠G. Let b be an element in G, but not in the cyclic subgroup generated by a.

O(a) = m and O(b) = n where O() refers tothe orders. . then how can we use this to construct a subgroup of G isomorphic to Zp×Zp?

Help will be appreciated. Thank you

Can you prove that if the prime factorization of |G| is ##p_1 p_2 p_3...p_n## and all of the primes ##p_i## are different, then G is cyclic? If two are the same, then what?
 
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  • #3
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If |G| =p1p2....pn and all of the primes are different, then |H| will be one of those primes. Since a group of prime order is cyclic, then H will be cyclic in that case . Infact all proper subgroups of G will be cyclic.
but i am not sure how to prove G as cyclic with this data..
 
  • #4
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If |G| =p1p2....pn and all of the primes are different, then |H| will be one of those primes. Since a group of prime order is cyclic, then H will be cyclic in that case . Infact all proper subgroups of G will be cyclic.
but i am not sure how to prove G as cyclic with this data..

What does the fundamental theorem of Abelian groups tell you?
 
  • #5
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the book which i am reading ( GAllian ) has not introduced this topic as of yet. In fact, not even normal and factor groups. i am on the chapter on external direct products.
 
  • #6
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the book which i am reading ( GAllian ) has not introduced this topic as of yet. In fact, not even normal and factor groups. i am on the chapter on external direct products.

Where exactly in Gallian is this?
 
  • #7
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Supplementary exercise for chapters 5 - 8 . Question no. 50 . Gallian 7/e contemporary guide to abstract algebra
 
  • #9
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OK. So let ##G## be finite abelian group such that ##G## does not contain a subgroup isomorphic to ##\mathbb{Z}_p\times \mathbb{Z}_p## for any prime. Consider the prime factorization ##|G|= p_1^{a_1} ... p_k^{a_k}:= p_1^{a_1}m##.

Define ##H = \{x\in G~\vert~\text{order of}~H~\text{divides}~p_1^{a_1}\}## and ##K = \{x\in G~\vert~\text{order of}~K~\text{divides}~m\}##.

Start by proving the following:
1) ##H## and ##K## are subgroups of ##G##.
2) For any ##x\in G## and integers ##s## and ##t##, we have ##x^{sp^k}\in H## and ##x^{tm}\in K##. Deduce that ##G=HK##.
3) Show that ##H\cap K = \{e\}##.
4) Find ##|H|## and ##|K|##.
5) Prove that ##H## is cyclic.
6) Prove that ##G## is the product of cyclic groups
7) Prove the result.
 
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  • #10
Dick
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OK. So let ##G## be finite abelian group such that ##G## does not contain a subgroup isomorphic to ##\mathbb{Z}_p\times \mathbb{Z}_p## for any prime. Consider the prime factorization ##|G|= p_1^{a_1} ... p_k^{a_k}:= p_1^{a_1}m##.

Define ##H = \{x\in G~\vert~\text{order of}~H~\text{divides}~p_1^{a_1}\}## and ##K = \{x\in G~\vert~\text{order of}~K~\text{divides}~m\}##.

Start by proving the following:
1) ##H## and ##K## are subgroups of ##G##.
2) For any ##x\in G## and integers ##s## and ##t##, we have ##x^{sp^k}\in H## and ##x^{tm}\in K##. Deduce that ##G=HK##.
3) Show that ##H\cap K = \{e\}##.
4) Find ##|H|## and ##|K|##.
5) Prove that ##H## is cyclic.
6) Prove that ##G## is the product of cyclic groups
7) Prove the result.

Your definition of K doesn't make much sense. I assume you mean ##K = \{x\in G~\vert~\text{order of}~x~\text{divides}~m\}##.
 
  • #11
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Your definition of K doesn't make much sense. I assume you mean ##K = \{x\in G~\vert~\text{order of}~x~\text{divides}~m\}##.

Yes, sorry!
 

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