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1.Prove that An¡ÁZ2 is not isomorphic to Sn when n¡Ý3. 2.Show that SOn¡ÁZ2 is not isomorphic to On when n is even. ×î½ü¿´½ü´ú£¬±íʾÕâÒ»ÀàÖ¤Ã÷²»Í¬¹¹µÄÌâ²»Ôõô»á°¡£¡ÇóÖ¸µ¼£¬×îºÃÊÇÓ¢ÎİæµÄ½â´ð£¡¶àлÁË£¡ [ ·¢×ÔÊÖ»ú°æ http://muchong.com/3g ] |
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zenmebuxing
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2Â¥2013-08-18 23:43:39
hank612
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˸ºÉØ»ª: ½ð±Ò+5, ¡ï¡ï¡ï¡ï¡ï×î¼Ñ´ð°¸ 2013-09-24 20:31:42
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you may use the information of center group to prove the first. It is known that Sn only has trivial center (when n>=3), this you can see from the conjugation action on the generating set (1k) k=2,...,n. (1k)^phi = (1^phi k^phi). but An X Z2 has nontrivial center, at least Z2 is in the center. (when n=3, the whole group is abelian). So no isomorphism. O_n (when n is even) has center I and -I. (you need to prove this, though I cannot come up a simple proof). but both these two elements belong to SOn. Therefore if On is isomorphic to SOn X Z2, then its center is at least (I, -I) X Z2, has four elements. which is not true. |
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3Â¥2013-08-19 00:32:47
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4Â¥2013-08-19 11:54:07
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5Â¥2013-08-19 12:03:28
hank612
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6Â¥2013-08-19 12:13:04
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7Â¥2013-08-19 13:25:28
hank612
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8Â¥2013-08-19 23:15:03
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9Â¥2013-08-20 19:29:27













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