Unformatted text preview: 258 5. ACTIONS OF GROUPS 5.4.3. Let P be a p –Sylow subgroup of a ﬁnite group G . Let H be a
subgroup of NG .P / such that jH j D p s . Show that H Â P . Hint: Refer
to Exercise 5.1.10, where it is shown that HP is a subgroup of NG .P /
with
jP j jH j
jHP j D
:
jH \ P j
5.4.4. Let p > q be prime numbers such that q divides p 1. Let ˛ and ˇ
be two injective homomorphisms of Zq into Aut.Zp / Š Zp 1 . Complete
the proof in Example 5.4.12 that Zp Ì Zq Š Zp Ì Zq .
˛ ˇ 5.4.5. Show that the groups Z30 , D15 , Z3 D5 , and Z5 D3 are mutually
nonisomorphic.
5.4.6. Verify the following isomorphisms:
(a) .Z5 Z3 / Ì Z2 Š D5 Z3
'1 (b) .Z5 Z3 / Ì Z2 Š Z5 (c) .Z5 Z3 / Ì Z2 Š D15 '2 D3 '3 5.4.7. Verify the assertion made about Aut.Z15 / in Example 5.4.14.
5.4.8. Show that an abelian group is the direct product of its p –Sylow
subgroups for primes p dividing jG j.
5.4.9. We have classiﬁed all groups of orders p , p 2 , and pq completely
(p and q primes). Which numbers less than 30 have prime decompositions
of the form p , p 2 , or pq ? For which n of the form pq does there exist a
nonabelian group of order n?
5.4.10. Let ˛ be the unique nontrivial homomorphism from Z4 onto hj i Â
Aut.Z7 /. Show that Z7 Ì Z4 is generated by elements a and b satisfying
˛ a7 D b 4 D 1 and bab 1 D a 1 , and conversely, a group generated by
elements a and b satisfying these relations is isomorphic to Z7 Ì Z4 .
˛ 5.4.11. Show that D14 and D7 Z2 are both groups of order 28 with 2–
Sylow subgroups isomorphic to Z2 Z2 . Give an explicit isomorphism
D14 Š D7 Z2 .
5.4.12. Is D2n isomorphic to Dn Z2 for all n? For all odd n? 5.4.13. Let N and A be groups, ˇ W A ! Aut.N / a homomorphism and
' 2 Aut.A/. Show that N Ì A Š N Ì A.
ˇ 5.4.14. Let W Z2 ˇ ı' Z2 ! Z2 be a surjective group homomorphism. ...
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 Fall '08
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 Algebra, Algebraic structure, Cyclic group, Group isomorphism, z2

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