1. The problem statement, all variables and given/known data
From Advanced Modern Algebra (Rotman):
Definition Let p be a prime. An abelian group G is p-primary if, for each ##a \in G##, there is ##n \geq 1## with ##p^na=0##. If we do not want to specify the prime p, we merely say that ##G## is primary .
If ##G## is any abelian group, then its p-primary component is ##G_p = \{a \in G : p^na=0 for some n\geq 1\}##.
I was trying to prove that ##G_p## is a subgroup.
2. Relevant equations
3. The attempt at a solution
If ##a## and ##b## are in ##G_p##, with ##p^ka=0## and ##p^na=0##, then ##p^{k+n}ab \in G_p##.
But now I need to show that if ##a \in G_p## then ##a^{-1} \in G_p##. If the operation is addition, then of course it is true, since ##\underbrace{a+...+a}_{p^k\text{ times}} = 0 \implies (-1)(\underbrace{a+...+a}_{p^k\text{ times}})=0##. But what if its any arbitrary operation?
From Advanced Modern Algebra (Rotman):
Definition Let p be a prime. An abelian group G is p-primary if, for each ##a \in G##, there is ##n \geq 1## with ##p^na=0##. If we do not want to specify the prime p, we merely say that ##G## is primary .
If ##G## is any abelian group, then its p-primary component is ##G_p = \{a \in G : p^na=0 for some n\geq 1\}##.
I was trying to prove that ##G_p## is a subgroup.
2. Relevant equations
3. The attempt at a solution
If ##a## and ##b## are in ##G_p##, with ##p^ka=0## and ##p^na=0##, then ##p^{k+n}ab \in G_p##.
But now I need to show that if ##a \in G_p## then ##a^{-1} \in G_p##. If the operation is addition, then of course it is true, since ##\underbrace{a+...+a}_{p^k\text{ times}} = 0 \implies (-1)(\underbrace{a+...+a}_{p^k\text{ times}})=0##. But what if its any arbitrary operation?
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