HELP: A finite abelian p-group generated by its elements of maximal order?

**Rita** · November 12th 2012, 10:38 PM

We say that a group in which every element has order a power of a fixed prime p is called a "p-group." Then how do I prove that a finite abelian p-group is generated by its elements of maximal order?

Algebra is so hard to understand. 囧rz

**jakncoke** · November 19th 2012, 03:59 AM

Assume G is a finite abelian p-group. Since G is finite, there exists $a \in G$ such that $|a| \geq |x|$ where $x \in G$ . This is a maximal element of G. Now if G = <a> = $p^n$ then we are done. Assume G $\not =$ <a> . Then $\exists a_1 \in G$ such that $a_1 \not \in <a>$ . Now since G is a group, closed under operation, $a_1 \dot a \in G$ so $a_1 \dot a$ has power $p^k \leq p^n$ , Now abelian groups has a property that for $(a_1 \dot a)^n = (a_1)^n \dot a^n$ . So using this property, $( a_1 \dot a )^{p^k} = (a_1)^{p^k} \dot a^{p^k} = e$ . If $(a_1)^{p^k} \not = e$ then it means $(a_1)^{p^k}$ is the inverse of $a^{p^k}$ , which would mean, since <a> is a subgroup, it contains its own inverses, so $(a_1)^{p^k} \in <a>$ which would mean $a_1 \in <a>$ which we said cannot be, so that means both $(a_1)^{p^k} = e$ and $a^{p^k} = e$ since $p^k \leq p^n$ , this means $p^k = p^n$ and $a_1$ is also a maximal element. Now you can see how this would proceed further, take an element not in $<a_1>$ or $<a>$ and u would come to the same conclusion about the new element and thus u can see that since G is finite, you get ${a, a_1, a_2, ..., a_k }$ are maximal elements which when intersected give u {e}, and which $<a> \cup <a_1> \cup ... <a_k> = G$ . Since all subgroups of an abelian group G, are normal in G. then indeed G = $<a> + <a_1> + <a_2> ... + <a_k>$

**Deveno** · November 19th 2012, 12:21 PM

Originally Posted by jakncoke

Assume G is a finite abelian p-group. Since G is finite, there exists $a \in G$ such that $|a| \geq |x|$ where $x \in G$ . This is a maximal element of G. Now if G = <a> = $p^n$ then we are done. Assume G $\not =$ <a> . Then $\exists a_1 \in G$ such that $a_1 \not \in <a>$ . Now since G is a group, closed under operation, $a_1 \dot a \in G$ so $a_1 \dot a$ has power $p^k \leq p^n$ , Now abelian groups has a property that for $(a_1 \dot a)^n = (a_1)^n \dot a^n$ . So using this property, $( a_1 \dot a )^{p^k} = (a_1)^{p^k} \dot a^{p^k} = e$ . If $(a_1)^{p^k} \not = e$ then it means $(a_1)^{p^k}$ is the inverse of $a^{p^k}$ , which would mean, since <a> is a subgroup, it contains its own inverses, so $(a_1)^{p^k} \in <a>$ which would mean $a_1 \in <a>$ which we said cannot be, so that means both $(a_1)^{p^k} = e$ and $a^{p^k} = e$ since $p^k \leq p^n$ , this means $p^k = p^n$ and $a_1$ is also a maximal element. Now you can see how this would proceed further, take an element not in $<a_1>$ or $<a>$ and u would come to the same conclusion about the new element and thus u can see that since G is finite, you get ${a, a_1, a_2, ..., a_k }$ are maximal elements which when intersected give u {e}, and which $<a> \cup <a_1> \cup ... <a_k> = G$ . Since all subgroups of an abelian group G, are normal in G. then indeed G = $<a> + <a_1> + <a_2> ... + <a_k>$

i think your argument has a flaw in it.

consider the abelian group Z₄xZ₈, which is a p-group with p = 2.

the element (1,1) is of order 8, which is the maximal possible order. note that:

<(1,1)> = {(0,0),(1,1),(2,2),(3,3),(0,4),(1,5),(2,6),(3,7)}

note that (3,1) is not in <(1,1)>. we have however that 4[(3,1)+(1,1)] = 4(0,2) = (0,0), with neither element being the identity, and 4 < 8.

two elements of order 8 may generate cyclic subgroups that intersect in a subgroup of order 4 (such as (1,1) and (3,1)) or order 2 (such as (1,1) and (0,1)). it is this difficulty i encountered when i started to post my own solution.

note that <(1,1),(3,1)> has 16 elements:

<(1,1),(3,1)> = {(0,0),(1,1),(2,2),(3,3),(0,4),(1,5),(2,6),(3,7),( 3,1),(1,3),(3,5),(1,7),(2,4),(0,2),(0,6),(2,0)}

if we consider <(1,1),(3,1),(0,1)>, we see that <(1,1),(3,1)> ∩ <(0,1)> has order 4: <(1,1),(3,1)> ∩ <(0,1)> = {(0,0),(0,2),(0,4),(0,6)}, which means that:

Z₄xZ₈ is generated by 3 elements of order 8: Z₄xZ₈ = <(1,1),(3,1),(0,1)>.

i think perhaps a counting argument (counting the number of elements of maximal order first) might be the way to proceed, but i haven't hit upon the right tactic yet.

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