Let G be a group with order pn where p is prime. Prove that any subgroup of order pn−1 is normal in G.
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Abstract Algebra:
Let G be a group with order pn where p is prime. Prove that any subgroup of order pn−1 is normal in G.
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- 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 2.27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .Prove or disprove that H={ [ 1a01 ]|a } is a normal subgroup of the special linear group SL(2,).
- Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?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.18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
- 9. Find all homomorphic images of the octic group.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 3.1.27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.