Please help me with my own proof for this: A group with finitely generated normal subgroup and finitely generated quotient is finitely generated itself
Let $G$ be a group with $G \trianglerighteq H$ normal subgroup. Assume that $H$ is finitely generated, and $G / H$ (the quotient group) is finitely generated as well.
Is $G$ finitely generated? (Answer: Yes.)
From finite generation we have $H=\langle h_1, …, h_n \rangle$ and $G/H = \langle g_1H,\ldots, g_m H \rangle$ for some $n,m \ge 1$. I've come up with my own proof based on the hints in previous question. I would like to please clarify this part:
From $G/H = \langle g_1H, \ldots, g_m H \rangle$, we have $G=\langle g_1, \ldots, g_m \rangle H$.
Questions:
- Is the following what is going on?
We are given: $G/H = \langle g_1 \bmod H, \ldots, g_m \bmod H \rangle$
Then we deduce: $G=\langle g_1,\ldots, g_m \rangle H$. This is now product set and not $\bmod$ or anything.
- So how do I prove (1) exactly? Here's what I tried:
Let $\pi_H$ be the canonical map ($\pi_H: G \to G/H, \pi_H(g)= gH$ as in $g \bmod H$).
$$\pi_H(G) = G/H = \langle g_1 \bmod H, \ldots, g_m \bmod H \rangle$$
$$\{(g_1 \bmod H)^{k_1} (g_2 \bmod H)^{k_2} \cdots (g_m \bmod H)^{k_m}\mid k_1, \ldots, k_m \in \mathbb Z\}$$
$$= \{(g_1^{k_1} \bmod H) (g_2^{k_2} \bmod H) \cdots (g_m^{k_m} \bmod H)|k_1, \ldots, k_m \in \mathbb Z\}$$
$$= \{g_1^{k_1}g_2^{k_2} \cdots g_m^{k_m} \bmod H\mid k_1, ldots, k_m \in \mathbb Z\}$$
$$= \{\pi_H(g_1^{k_1}g_2^{k_2} \cdots g_m^{k_m})\mid k_1, \ldots, k_m \in \mathbb Z\}$$
$$= \pi_H(\{(g_1^{k_1}g_2^{k_2} \cdots g_m^{k_m}\mid k_1, \ldots, k_m \in \mathbb Z\})$$
$$= \pi_H(\langle g_1,\ldots, g_m \rangle)$$
And then
$$\pi_H(\langle g_1, \ldots, g_m \rangle) = \{g \bmod H \mid g \in \langle g_1, \ldots, g_m \rangle \}$$
while
$$\pi_H(G) = G/H = \{g \bmod H | g \in G \}$$
Not sure how to conclude from here. Is there some rule like for elements $a,b \in G$, we get from $\pi_H(a) = \pi_H(b)$ that $a \bmod H = b \bmod H$ but for subsets/subgroups $A,B \subseteq G$, we get from $\pi_H(A) = \pi_H(B)$ that $AH=BH$? (well perhaps on the element level we get $\{a\}H=\{b\}H$)
- Edit 1: Got it I think. See an answer I posted. Actually I made mistakes above
- in saying that
$$\langle g_1 \bmod H, \ldots, g_m \mod H \rangle = \{(g_1 \bmod H)^{k_1} (g_2 \bmod H)^{k_2} \cdots (g_m \bmod H)^{k_m}\mid k_1, \ldots, k_m \in \mathbb Z\}$$
- in assuming that $$\{(g_1^{k_1}g_2^{k_2} \cdots g_m^{k_m}\mid k_1, \ldots, k_m \in \mathbb Z\}$$ (when I was saying that $$\pi_H(\{(g_1^{k_1}g_2^{k_2}\cdots g_m^{k_m}\mid k_1, \ldots, k_m \in \mathbb Z\}) = \pi_H(\langle g_1, \ldots, g_m \rangle)$$).
- Is it nonsensical to continue the argument with something like the following?
From $G/H = \langle g_1H,\ldots, g_m H \rangle$, we have $G=\langle g_1, \ldots, g_m \rangle H$.
Then
$$G=\langle g_1, \ldots, g_m \rangle H$$
$$=\langle g_1, \ldots, g_m \rangle \langle h_1, \ldots, h_n \rangle$$
$=$ something like $\langle g_1, \ldots, g_m, h_1, \ldots, h_n \rangle$ or $\langle \{g_ih_j \mid i=1,\ldots,m; j=1,\ldots, n\} \rangle$
- Edit 2: I think it's sensible because the product set $\langle g_1, \ldots, g_m \rangle \langle h_1, \ldots, h_n \rangle$ is given to be equal to a subgroup of $G$ (namely the whole of $G$) and thus we can just combine the indices/generators.
Best Answer
Wait after overcoming my fear of using $gH$ when what is meant is $g \mod H$, I think I figured it out. Might as well just type it here as an answer.
To prove $G = \langle g_1, \ldots, g_m\rangle H$:
$\supseteq$ duh
$\subseteq$ Let $g \in G.$ Then $gH = \pi_H(g) \in \pi_H(G)=\pi_H(\langle g_1, \ldots, g_m \rangle) = \{kH\mid k \in \langle g_1, \ldots, g_m \rangle \}$. Then $gH=kH$, for some $k \in \langle g_1, \ldots, g_m \rangle$. Hence, $gh=kl$, for some $h,l \in H$. Therefore, $g = \underbrace{k}_{\in \langle g_1, \ldots, g_m \rangle}\underbrace{lh^{-1}}_{\in H}$
Wait I also figured out $AH=BH$ (given $\pi_H(A) = \{a \bmod H\mid a \in A\} = \{b \bmod H\mid b \in B\} = \pi_H(B)$) even if you use $\bmod H$:
$\supseteq$ By symmetry, 1 direction is sufficient.
$\subseteq$ Let $ah \in AH$, for $(a,h) \in A \times H$. We must show that $ah=bk$, for some (b,k) \in B \times H.
Now, when $a \bmod H = \pi_H(a) \in \pi_H(A) = \pi_H(B) = \{b \bmod H\mid b \in B\}$. Then $a \bmod H = c \bmod H$, for some $c \in B$. Hence, $ca^{-1}=q$, for some $q \in H$. Therefore, $ah=cq^{-1}h$. Choose $(b,k)=(c,q^{-1}h)$. (or simply note that $ah=\underbrace{c}_{\in B}\underbrace{q^{-1}h}_{\in H} \in BH$.)