Subgroups of a nilpotent group are nilpotent

[oblixps]

i know that this can be proved easily using the lower central series, but i am having a hard time trying to prove this using the upper central series definition.

I saw a proof which said that if H is a subgroup of G, $$Z_{r}(H) \geq Z_{r}(G) \cap H$$ for all r. So if G is nilpotent which means that $$Z_{n}(G) = G$$ for some G, then this implies that $$Z_{n}(H) = H$$ for some n and therefore H is nilpotent.

i am having trouble understanding why $$Z_{r}(H) \geq Z_{r}(G) \cap H$$ for all r. I can see that the result is true for r = 1 since we are dealing only with the centers of G and H, but i am having trouble proving the induction step in order to prove this for all r. Can someone give me some hints on this?

 

[jvicens]

First of all, let's define what the upper central series and lower central series are. The lower central series of a group G is defined as a sequence of subgroups of G, denoted as G_i, where G_1 = G and G_i+1 = [G,G_i], the commutator subgroup of G and G_i. The upper central series is defined as a sequence of subgroups of G, denoted as Z_i, where Z_1 = {e}, the trivial subgroup, and Z_i+1 = Z(G_i), the center of G_i.

Now, let's look at the statement Z_r(H) ≥ Z_r(G) ∩ H for all r. This means that the center of the rth term in the lower central series of H is greater than or equal to the intersection of the center of the rth term in the lower central series of G and H. In other words, the center of H_i is greater than or equal to the intersection of the center of G_i and H for all i ≤ r.

To prove this statement, we will use induction on r. The base case, r = 1, is trivial since we are only dealing with the centers of G and H. Now, let's assume that the statement is true for some r = k, i.e. Z_k(H) ≥ Z_k(G) ∩ H. We want to show that this implies the statement is true for r = k+1.

From the definition of the lower central series, we know that G_k+1 = [G,G_k] and H_k+1 = [H,H_k]. So, any element in G_k+1 can be written as a product of commutators of elements in G and any element in H_k+1 can be written as a product of commutators of elements in H. Now, since Z_k(G) and Z_k(H) are subgroups of G_k and H_k respectively, any element in their intersection, Z_k(G) ∩ H, can be written as a product of commutators of elements in both Z_k(G) and H.

Therefore, any element in G_k+1 ∩ H_k+1 can be written as a product of commutators of elements in G and H. This means that G_k+1 ∩ H_k+1 is a subgroup of [G,H], the commutator subgroup