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### If $G$ is an infinite group, and $A,B$ subgroups of finite index in $G$, then $A \cap B$ has finite index in $G$ [duplicate]

Question: If $G$ is an infinite group, and $A, B$ subgroups of finite index in $G$, then prove $A \cap B$ has finite index in $G$. I'm trying to show that $A\cap B$ can not have infinite index, but I ...
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### A question on Cosets [duplicate]

Let $G$ be a group and $H$ , $K$ be subgroups of $G$ such that $[G:H]$ and $[G:K]$ are finite. Then is it true that $[G:H∩K]$ is also finite ?
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### $[G : H] < ∞$ and$[G : K] < ∞$ then $[G : H ∩ K] < ∞$ where $H$ and $K$ be subgroups of $G$ [duplicate]

Let $H$ and $K$ be subgroups of a group $G$. Then is the following true ? If $[G : H] < ∞$ and $[G : K] < ∞$, then $[G : H ∩ K] < ∞$. I think this's false because there's still a case ...
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### Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.

Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$. I need this theorem to prove something.
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### A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$

I would be very thankful if someone could give me a hint with proving this. It's a very common exercise in abstract algebra textbooks. If $G$ is a group with a subgroup $H$ of finite index $n$, ...
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### When are finite-index subgroups of a Lie group closed?

Let $G$ be a Lie group (or, if necessary, a reductive Lie group) and $H$ a subgroup of $G$. If $\lbrack G:H\rbrack < \infty$, is it true that $H$ is closed? If not, are there any broad assumptions ...
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### If $[G:H]<\infty$, then $H$ contains a normal subgroup $N$ of $G$ such that $[G:N]<\infty$.

Let $G$ be a group and $H$ be a subgroup of $G$. I want to prove that if $[G:H]<\infty$, then $H$ contains a normal subgroup $N$ of $G$ such that $[G:N]<\infty$. Professor gave me the following ...
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### Bounding the index of subgroup intersection

Full disclosure: this is a homework problem, but it is not assigned to turn in for credit. The problem is from Dummit and Foote, Chapter 3.2: Suppose $H, K$ are subgroups of finite index a group (not ...
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### Index of intersection of finite index subgroups

I'm trying to understand this proof https://math.stackexchange.com/a/128549/205193 which does not seem complicated but I don't understand why : why $p(x)=p(y)$, implies that $x$ and $y$ are in the ...
### On formula $\sum_{i=1}^n 1/(G : H_i) = 1$ on a group $G$
Let $G$ be a group. Let $H$ be a subgroup of $G$ such that $(G : H) \lt \infty$. Then there exists a sequence of elements $a_1,\cdots, a_n$ such that $G = \bigcup_{i=1}^n a_iH$ is a disjoint union. ...
### If $H$ and $K$ are subgroups of $G$, what does the notation $HK$ mean?
I know there is direct product $H \times K$ and semidirect product, but what is implied when it is just $HK$ without any symbols in between? For example, in this question it says $[HK:K] \leq [G:K]$. ...