Typo in Algebra by Artin regarding center of the special linear group $SL_n(\mathbb R)$

Question

I will link the photo in question here:

Shouldn't the last line read, "$SL_n(\mathbb R)$ consists of $I, -I$ if n is even, and is trivial if n is odd"?

So the is not a typo in Algebra by Artin (see title). – Dietrich Burde Dec 09 '19 at 20:27 — Dietrich Burde, Dec 09 '19 at 20:27

score 3 · Accepted Answer · answered Dec 09 '19 at 14:12

3

What is written is correct. The center of $SL_2(\mathbb{R})$ consists of $I,-I$. The author does not talk about $SL_n(\mathbb{R})$ for $n \geq 3$.

But you are right, for $n$ odd $-I$ has determinant $-1$ and thus does not live in $SL_n(\mathbb{R})$, thus definitely also not in its center.

answered Dec 09 '19 at 14:12

J. De Ro

The author does talk about what happens for $n>3$ in the last line. – Sally G Dec 09 '19 at 14:17
Am I right about $n$ being even, too, since $-1^{2m}=1$? – Sally G Dec 09 '19 at 14:18
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The author says that the center of $S_n$ is trivial for $n \geq 3$. $S_n$ is another group than $SL_n(\mathbb{R})$ – J. De Ro Dec 09 '19 at 14:18

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