Remark: If (H, ) and (K,) are subgroup of a group (G, ) there fore (HUK, ) need not be a subgroup of a group (G,) this can be explained by the following example

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

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Q: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…

A: Q1.  Third option is correct. Q2. Second option is correct.

Q: For any subset H of a group G, if ab^-1 is in H for all a,b element of H then H is a subgroup of G.…

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Q: Suppose that p:G→G'is a group homomorphism. Show that () p(e) = ¢(e') (1) For every gEG, (ø(g))-l =…

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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: 32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same…

A: To show:

Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: 4) Let H and K be a subgroup of a group G if HAK ,KAG and HAG then HnK is not normal subgroup of G.…

A: We have to use normal subgroup definition to solve given problem. We know that A subgroup H<G is…

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: 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 with |G|=187 then every proper subgroup of G is:

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Q: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

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: 4) Let H and K be a subgroup of a group Gif HAK,KAG and HAG then HnK is not normal subgroup of G.…

A: Given, H and K are subgroups of a group G. Also, given that H is normal in G and K is normal in G.…

Q: Question 2. Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G…

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Q: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

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Q: {hk | h ∈ H, k ∈ K}}

A: We have to prove that  {hk|h∈H, k∈K} is a subgroup of G.

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: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

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

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Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

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Q: Theorem(7.9): If (H, *) is a subgroup of the group (G, *). then Va e G the pair (a+H a,+) is a…

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Q: (a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…

A: PART A: Since, Every subgroup is a group in itself. To prove &lt;g&gt; is a subgroup ,One step…

Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

A:

Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10…

A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…

Q: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Consider the finite simple group G that has subgroup H and K. |G: H| and |G: K| are relatively…

Q: The centralizer and normalizer of a subgroup of a group are the same . If thats true give deatiled…

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Q: Let H and K be two subgroups of a group G. Let HK={ab|a∈H,b∈K}. Then HK is a subgroup of G. true or…

A: F hv

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: The identity element in a subgroup H of a group G must be the same as the identity element in G…

A: The identity element in a subgroup H of a group G must be the same as the identity element in G.

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: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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Q: a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…

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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: If H and K are subgroups of a group G then H n K is a subgroup of G.

A: Note:  according to our guidelines we can answer first question and rest can be reposted.   Lemma:…

Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N

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Q: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False

A: we will give the counter example in support of our answer.

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

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Q: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…

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Q: (a) If G is abelian and A and B are subgroups of G, prove that AB is a subgroup of G. (b) Give an…

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

A:

Q: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

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Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups that G has is 4 O 12 O 18…

A: Given order of G is 36 So U(G) = {1,5,7,11,13,17,19,23,25,29,31,35} So number of elements are 12…

Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups %3D that G has is 4 О 12 О…

A: Order of a group: Let G be a group and n be the number of elements in the group. Then, order of…

Q: Suppose that | G|= 20. What're possible orders for a subgroup of G. 35.

A: We can find the subgroup of G as below.