Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal subgroup of order 28.

Q: Suppose that a group G of order 231 has a normal subgroup N of order 11. Then, G/N is cyclic O False…

A: Given that G is a group of order 231 and N is an normal sub-group of G of order 11. To show: G/N is…

Q: Let (G,*) be an a belian group, if (H,) and (K,*) are subgroup of (G,*) then (H * K,*) is a subgroup…

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Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: Let G be a group of odd order, and let N be a normal subgroup of order 5. Show that N is contained…

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Q: Let N be a normal subgroup of G and let K/N be a normal subgroupof G/N. Prove that K is a normal…

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Q: A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is…

A: The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If we…

Q: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

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Q: Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that NG(H)…

A: " Let G be a group and H ≤ G.The subgroup H is normal in its normalizer NG(H), this imply that NG(H)…

Q: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

A: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

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Q: Compute the Cayley table and find a familiar isomorphic group of G/N where G is a group and N is any…

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Q: Let G be a group of order 100 that has a subgroup H of order 25.Prove that every element of G of…

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Q: 51. Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove…

A: According to our guidelines we can answer only first question and rest can be reposted. Not more…

Q: 12. Prove that the intersection of any family of normal subgroups of a group (G, *) is again normal…

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Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…

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Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…

A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…

Q: . Let H and K be normal subgroups of a group G such that HCK, show that K/H is a normal subgroup of…

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Q: Let G be a group, prove that the center Z(G) of a group G is a normal subgroup of G.

A: Let G be a group. Consider the subgroup ZG=x∈G | ax=xa.

Q: Let G be a group and let H be a subgroup of G with |G : H| = 2. Prove that H a G, that is, H is a…

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Q: . Let H and K be normal subgroups of a group G such nat HCK, show that K/H is a normal subgroup of…

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Q: In Z24, list all generators for the subgroup of order 8. Let G = <a>and let |a| = 24. List all…

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Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

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Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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Q: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

Q: Let H and K be normal subgroups of a group G such at HCK, show that K/H is a normal subgroup of G/H.

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Q: How thata Show that an intersection of normal subgroups of a group G is again a normal subgroup of…

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Q: Let G be a group of order 90. show that G has at most one subgroup of order 45

A: Given: G be a group of order 90

Q: Show that every group of order (35)3 has a normal subgroup of order 125

A: Given, A Sylow 5-subgroup of a group of order 353 is of order 125. The divisors of 353 that are not…

Q: Prove that if H is a normal subgroup of G of prime index p. (Note G can be finite or infinite…

A: It is given that, H is a normal subgroup of G of prime index p. (Here G can be a finite or infinite…

Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: If G is a group with 8 elements in it, and H is a subgroup of G with 2 elements, then the index…

A: We are provided that a group G with 8 elements and H is a subgroup of G with 2 elements and…

Q: Let H and K be normal subgroups of a group G such that HCK, show that K/H is a normal subgroup of…

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Q: 2. A Sylow 3-subgroup of a group of order 54 has order

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Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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Q: Prove that any subgroup H (of a group G) that has index 2 (i.e. only 2 cosets) must be normal in G

A: To show that H is a normal subgroup we have to show that every left coset is also a right coset. We…

Q: The Kernal of any group homomorphism is normal subgroup True False

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Q: If H is a subgroup of G such that [G : H] = 2, then show that H is a normal subgroup of G.

A: Suppose H≤G such that [G:H] = 2. Thus H has two left cosets (and two right cosets) in G.

Q: Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup…

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Q: G is a finite group of order IGI =pqr with p< q <r prime

A: Solution

Q: Let G be a finite group and let H be a normal subgroup of G. Provethat the order of the element gH…

A: Given: G be a finite group and H be a normal subgroup of G.

Q: think of this as being a stronger type of normality. Prove that a characteristic subgroup is normal…

A: A subgroup H of h is called normal subgroup of h if θH⊆H  ∀θ∈AutG

Q: Let let G₁ be A be of Suppose Subgroup index a group and a normal of finite G+₁ that H

A: We know that if G is a group and H is a subgroup of G and x is an element in G of finite order n. If…

Q: 5. Show that the intersection of two normal subgroups of G is a normal subgroup of G 6. If G is a…

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Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

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

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Q: Let G be a group of order 24. Suppose that G has precisely one subgroup of order 3, and one subgroup…

A: Theorem : If a group G is the internal direct product of subgroups H and K, then G is isomorphic to…