If G is a group with 8 elements in it, and H is a subgroup of G with 2 elements, then the index [G:H]=4. How can I tell if H is a normal subgroup just based on that information?

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: Let G be a group and H, K are subgroups of G with HK=KH. Prove that HK is a subgroup of G.

A: Given that, G be a group and H, K are sub groups of G with HK=KH. Let x∈HK. Then x=hk for some…

Q: Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the…

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Q: Prove that if N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal…

A: To Prove If N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal subgroup…

Q: 11. Find the cyclic subgroup of D4 generated by µp². What is the order of this subgroup?

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Q: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

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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: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

Q: Theorem: Let (K,+)is a subgroup of a group (H, ) and (H,) is a subgroup of a group (G,) then (K, )is…

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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: Suppose H is a distant and normal subgroup of a group G. Prove that each subgroup of H is a normal…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…

A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.

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

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Q: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.

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Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n,…

A: Given, G is a finite group and H is a subgroup of G of order n.

Q: Let G be a finite p-group of order p". Show that for all 0<kSn, there is a subgroup order p and each…

A: Given: Let G be a finite p-group of order pn. We have to prove for all 0≤k≤n there is a subgroup of…

Q: 17. Show that every group of order (35)° has a normal subgroup of order 125.

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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 H and K are two subgroups of finite indices in G, then show that H ∩ K is also of finite index in…

A: If H and K are two subgroups of finite indices in G, then show that H ∩ K isalso of finite index in…

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 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: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

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: The subgroup {e} is called the nontrivial, that is, a subgroup that is not e is nontrivial.…

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Q: If H and K are normal subgroups of G, show that their intersection is also a normal subgroup. To do…

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Q: Question 7. (10 Marks) If K is a subgroup of G and N is a normal subgroup of G, prove that KnN is a…

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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: 6. (b) For each normal subgroup H of Dg, find the isomorphism type of its corresponding quotient…

A: First consider the trivial normal subgroup D8. The quotient group D8D8=D8 and hence it is isomorphic…

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

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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: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

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Q: If H and K are subgroups of G, Show tht H intersecting with K is a subgroup of G.  (Can you see that…

A: Use the 2-step subgroup test to prove H Ո K is a subgroup, which states that,

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

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Q: What is the relationship between a Sylow 2-subgroup of S4 and the symmetry group of the square? that…

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Q: H. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.

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Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

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Q: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

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Q: For A, the alternating subgroup of S, show that it is a normal subgroup, write out the cosests, then…

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Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

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Q: If H is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…

A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.

Q: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

Q: Prove that a group of order 595 has a normal Sylow 17-subgroup.

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Q: Prove that every group of order 78 has a normal subgroup of order 39.

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Q: Let G be a group and let a E Ga G with a = 8. the order of a² is not equal to the order of the…

A: The given statement is

Q: 5. If H. aEA are a family of subgroups of the group G, show that is a subgroup of G.

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Q: 1. Let G be a group and let H, H, .. H, be the subgroups of G. The ...

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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…