# Definition of the Tate group associated with $p$-divisible groups

Let $A$ be an abelian group, $p$ a prime number and $p_A:A \rightarrow A$ the multiplication by $p$. Let $A\left[p^n\right]$ be the kernel of $p_A$ composed $n$ times with itself. Then Lang, in his algebra p. 50, defines the Tate group $T_p(A)$ associated with the $p$-divisible group $A$ as the inverse limit of the inversely directed family $\left(A\left[p^{n+1}\right]\right)$.

As i understand, $n=0,1,...$ By definition, for $n=0$, the corresponding component of any element of $T_p(A)$ is zero and so for $n=1$ the component of any element of $T_p(A)$ must be inside $A\left[p\right]$. But substituting for $n=1$ at the defining formula for $T_p(A)$ yields $A\left[p^2\right]$.

Can anyone advise me about the correct interpretation of the definition? Thanks :-)

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