Abstract Algebra: Prove that if p is a prime and G is a group of order p α for some α ∈ Z +, then every subgroup of index p is normal in G.
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Q: B. Let G be a group of order 60. Is there exist a subgroup of G of order 24? Explain your answer
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Abstract Algebra:
Prove that if p is a prime and G is a group of order p α for some α ∈ Z +, then every subgroup of index p is normal in G.
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- 9. Let be a group of all nonzero real numbers under multiplication. Find a subset of that is closed under multiplication but is not a subgroup of .True or False Label the following statements as either true or false. 1. Every finite group of order is isomorphic to a subgroup of order of the group of all permutations on .27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .
- True or False Label each of the following statements as either true or false. Let H be a subgroup of a finite group G. The index of H in G must divide the order of G.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.Let G be a group of finite order n. Prove that an=e for all a in G.
- True or False Label each of the following statements as either true or false. If a group G contains a normal subgroup, then every subgroup of G must be normal.Exercises Find an isomorphism from the octic group D4 in Example 12 of this section to the group G=I2,R,R2,R3,H,D,V,T in Exercise 36 of Section 3.1.Prove that SL(2,R)={ [ abcd ]|adbc=1 } is a subgroup of GL(2,R), the general linear group of order 2 over R.The subgroup SL(2,R) is called Special linear group of order 2 over R.
- Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.True or False Label each of the following statements as either true or false. Every normal subgroup of a group is the kernel of a homomorphism.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.