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## Main Question or Discussion Point

Having trouble with a couple of algebra questions and would really appreciate any hints or pointers.

1. A is a subgroup of group G with a finite index. Show that

[tex]N = \bigcap_{x \in G}x^{-1}Ax[/tex]

is a normal subgroup of finite index in G.

I'm able to show that N is a subgroup of G by applying the subgroup test. Thing is, I'm not sure how to prove that it's a normal subgroup. It seems that the fact that A is of finite index should play into it somehow.

2. Let [tex]G = GL(n,\mathbb{Z})[/tex] for [tex]n \ge 2[/tex]. Define the n-th converge subgroup, G(m), as [tex]G(m) = \left\{A \in G : A\equiv I_n\mod m\right\}[/tex].

Show that G(m) is a normal subgroup.

Tried thinking of this as x^-1yx where x is just a GL matrix and y is one of G(m) and trying to show that this product is one of G(m). Wrote some formulas for individual entries of the product matrix, but doesn't seem to work in terms of guaranteeing that each non-diagonal entry is a multiple of m, and every digonal entry is a multiple of m and with an extra 1. Although perhaps it's just that this gets somewhat messy and I made some silly mistake somewhere.

Thanks in advance. Any help is really appreciated.

1. A is a subgroup of group G with a finite index. Show that

[tex]N = \bigcap_{x \in G}x^{-1}Ax[/tex]

is a normal subgroup of finite index in G.

I'm able to show that N is a subgroup of G by applying the subgroup test. Thing is, I'm not sure how to prove that it's a normal subgroup. It seems that the fact that A is of finite index should play into it somehow.

2. Let [tex]G = GL(n,\mathbb{Z})[/tex] for [tex]n \ge 2[/tex]. Define the n-th converge subgroup, G(m), as [tex]G(m) = \left\{A \in G : A\equiv I_n\mod m\right\}[/tex].

Show that G(m) is a normal subgroup.

Tried thinking of this as x^-1yx where x is just a GL matrix and y is one of G(m) and trying to show that this product is one of G(m). Wrote some formulas for individual entries of the product matrix, but doesn't seem to work in terms of guaranteeing that each non-diagonal entry is a multiple of m, and every digonal entry is a multiple of m and with an extra 1. Although perhaps it's just that this gets somewhat messy and I made some silly mistake somewhere.

Thanks in advance. Any help is really appreciated.

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