6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.

Q: 2.26. If G is a group and H is a subgroup of index 2 in G, then prove that H is a normal subgroup of…

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Q: 3. Consider the group S. a) For the subgroup H= {(1),(13)} , write all cosets that can be formed. b)…

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Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

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Q: Every

A: We will be using sylow's theorems and it's consequences to arrive at the conclusion that statement…

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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Q: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

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Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: a) Show that a group of order 700 has a normal subgroup of order 175.

A: To show that a group of order 700 has a normal subgroup of order 175, we will make use of the Sylow…

Q: 10. Let G be abelian and let H be a subgroup of G. Show that G/H is abelian.

A: We have to prove that given theorem.

Q: Let G be a group of order 30. Does G have to have a subgroup of order15? Does G have to have an…

A: Group of order 30 does not necessarily have a subgroup of order 15, nor does it necessarily have an…

Q: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

Q: Let G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

Q: Let K be a subgroup of G and let H = {g : gKg^-1} = K. Show that H is a subgroup of G

A: Let K be a subgroup of G and H be defined as H = { g : gKg-1 = K }. Then, we have to show that H is…

Q: 9. Let G and H be two groups, and G = {(9, en) | g = G}. Prove that G is a normal subgroup of G x H,…

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Q: 4. Let G be a group of order 49. Prove that G is cyclic or g7 = e for all g € G.

A: Given a group of order 49, then to prove that either G is cyclic or g7=e for all g∈G.

Q: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

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Q: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

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Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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Q: Draw the lattice of the subgroups Z/20Z.

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Q: 3. Let G be a group of order pq where p, q are two distinct prime numbers. (a) Assuming that p < q…

A: Given a group G of order pq where p and q are two distinct prime numbers.   Used definition:  number…

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Prove that If |G|=np with p>n and p prime, and H is a subgroup of G with order p, then: H is normal…

A: We proved by contradiction that H is normal in G  The procedure is given below

Q: 4. Let G be a group and let H, K be subgroups of G such that |H| = 12 and |K| = 5. Prove that HNK =…

A: We have to prove given result:

Q: . Let G be a group of order 21. Show that the number of order 3 subgroups of G is either one or…

A: Given that let G be a group of order .To prove:- The number of order  subgroups of G is either one…

Q: Let G be a group of order p™ where p is a prime number and m is a positive integer. Show that G…

A: given G be a group of order pm where p is a prime number and m is a positive integer to show G…

Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

A: It is given that, H be a subgroup of S5 which contains a 5-cycle and a 2-cycle. We have to show…