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
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
Assume G is a finite abelian p-group. Since G is finite, there exists such that where . This is a maximal element of G. Now if G = <a> = then we are done. Assume G <a> . Then such that . Now since G is a group, closed under operation, so has power , Now abelian groups has a property that for . So using this property, . If then it means is the inverse of , which would mean, since <a> is a subgroup, it contains its own inverses, so which would mean which we said cannot be, so that means both and since , this means and is also a maximal element. Now you can see how this would proceed further, take an element not in or and u would come to the same conclusion about the new element and thus u can see that since G is finite, you get are maximal elements which when intersected give u {e}, and which . Since all subgroups of an abelian group G, are normal in G. then indeed G =
i think your argument has a flaw in it.
consider the abelian group Z_{4}xZ_{8}, 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:
Z_{4}xZ_{8} is generated by 3 elements of order 8: Z_{4}xZ_{8} = <(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.