homework4

# homework4 - the dodecahedron to the icosahedron and the the...

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Math 113 Section 5 Homework #4 Fall 2010 Due: Monday, October 4 1. Let X be a set on which a group G acts. Show that the kernel of the action and the stabilizer G x of an element x X are subgroups of G . 2. Section 2.2, Exercise 6: Let H be a subgroup of a group G . Let N G ( H ) and C G ( H ) be the normalizer and centralizer of H , respectively. (a) Show that H N G ( H ) . Give an example to show that this is not necessarily true if H is not a subgroup. (b) Show that H C G ( H ) if and only if H is abelian. 3. Section 2.2, Exercises 21 and 22: (a) Show that the group of rotations of a cube is isomorphic to S 4 . (This group acts on the set of 4 pairs of opposite vertices.) (b) Show that the group of rotations of an octahedron is also isomorphic to S 4 . (This group acts on the set of 4 pairs of opposite faces.) (Notice that you have shown that the groups of rotations of these two “Platonic solids” are isomorphic. This is because the solids are somehow “dual” to one another. It’s pretty fun to visualize this geometric duality as well as the analogous dualities relating

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Unformatted text preview: the dodecahedron to the icosahedron and the the tetrahedron to itself.) 4. Let G = é g ê with | G | = n and G Í = é g Í ê with | G Í | = m . Show that G × G Í is cyclic if and only if ( m,n ) = 1 . 5. Let G be a ﬁnite cyclic group with | G | = p where p is prime. Show that G has no proper subgroups. 6. Show that Q × Q is not cyclic. 7. Section 2.4, Exercise 2: Prove that if A is a subset of B then é A ê ≤ é B ê . Give an example where A ⊂ B with A Ó = B but é A ê ≤ é B ê . 1 8. Let G be a group, x ∈ G and y ∈ C G ( x ) . Show that é x,y ê is abelian. 9. Section 2.3, Exercise 21: Let p be an odd prime and n be a positive integer. Use the Binomial Theorem to show that (1 + p ) p n-1 ≡ 1 (mod p n ) but (1 + p ) p n-2 is not equivalent to 1 (mod p n ). Dedude that 1 + p is an element of order p n-1 in the multiplicative group ( Z /p Z ) × . 2...
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## This note was uploaded on 01/21/2012 for the course MATH 113 taught by Professor Ogus during the Fall '08 term at Berkeley.

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homework4 - the dodecahedron to the icosahedron and the the...

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