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College Algebra Exam Review 247

College Algebra Exam Review 247 - 5.4 GROUP ACTIONS AND...

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5.4. GROUP ACTIONS AND GROUP STRUCTURE 257 and NA D G , because j NA j D j N jj A j j N \ A j D 28 . Thus G is the semidirect product of N and A . The abelian groups of order 28 are Z 7 Z 4 and Z 7 Z 2 Z 2 . To classify the non-abelian groups of order 28, we have to classify the non- trivial homomorphisms from groups of order 4 into Aut . Z 7 / Š Z 6 . Aut . Z 7 / has a unique subgroup of order 2, generated by the automor- phism j W OExŁ 7 7! OE 7 . Any non-trivial homomorphism from a group of order 4 into Aut . Z 7 / must have image h j i , since the size of the image is a common divisor of 4 and 6. So we are looking for homomorphisms from a group of order 4 onto h j i Š Z 2 . Z 4 has a unique homomorphism ˛ onto h j i determined by ˛ W OE1Ł 4 7! j . Therefore, up to isomorphism, Z 7 Ì ˛ Z 4 is the unique non-abelian group of order 28 with 2 –Sylow subgroup isomorphic to Z 4 . This group is gen- erated by elements a and b satisfying a 7 D b 4 D 1 and bab 1 D a 1 .
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