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Are the matrices which are subset of $GL_n(\mathbb{R})$,$\begin{pmatrix} 1&&a_{12}\\0&&a_{22}\end{pmatrix}$ and $\begin{pmatrix} a_{11}&&a\\0&&a\end{pmatrix}$ normal subgroups of $GL_2(\mathbb{R})$?

I tried finding the conjugate of any general invertible matrix with the above matrices but did not get an upper triangular matrix in the end. Any help. Thanks beforehand.

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  • $\begingroup$ You're correct that conjugation will sometimes not yield an upper triangular matrix in the end. Since those subgroups are not closed under conjugation (note that the second is not a subgroup unless you ignore the zero matrix), they are not normal subgroups. It suffices to find one counter-example, and often examples are helpful for understanding a problem anyway. There is no reason to compute a general conjugate when a specific, easily computable example will probably work. $\endgroup$ – Hans Musgrave Jan 5 '17 at 7:02
  • $\begingroup$ @user3798897 sorry, modified the post. $\endgroup$ – vidyarthi Jan 5 '17 at 7:04
  • $\begingroup$ That second set still has the zero matrix. I'm less sure it forms a subgroup anyway though. In fact I'm nearly certain it doesn't. $\endgroup$ – Hans Musgrave Jan 5 '17 at 7:07
  • $\begingroup$ @user3798897 ok, what if the sets are subsets of the group $GL_2(\mathbb{R})$? $\endgroup$ – vidyarthi Jan 5 '17 at 7:11
  • $\begingroup$ They are definitely subsets, but to be a normal subgroup (is that what you want?) you need for the sets to be subgroups which are closed under conjugation. Neither of them are since you can take 'most any matrix in $GL_2(\mathbb{R})$ with positive elements and conjugate 'most any matrix in your sets and get something which is not upper triangular, so definitely not in your sets. $\endgroup$ – Hans Musgrave Jan 5 '17 at 7:14
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You definitely have subsets, but to be a normal subgroup you need for the sets to be subgroups which are closed under conjugation. Neither of them are since you can take 'most any matrix in $GL_2(\mathbb{R})$ with positive elements and conjugate 'most any matrix in your sets and get something which is not upper triangular, so definitely not in your sets.

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