Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с. 2 O d. 4 O e. 1

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

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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: 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: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

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Q: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

Q: If d divides the order of a cyclic group then this group has a subgroup of order d. Birini seçin: O…

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Q: Given that G is a group and H is a subgroup. What is the result of (b^-1)^-1 if b is an element of…

A: Given that     G is a group and H is a subgroup of G.    Inverse of an element:  Let G be a group…

Q: is a subgroup of Z1, of order: 3 12 O 1 The following is a Cayley table for a group G. 2. 3.4 = 2 3…

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Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: G let then. [b, a]= be an group and Ta %3D

A: Given that G is a group and also a,b,c∈G. To prove that b,a= a,b-1 Since G is a group, it satisfies…

Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

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Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

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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: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

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Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, Cg(A) = A and…

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Q: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

Q: Let E = Q(√2, √5). What is the order of the group Gal(E/Q)?What is the order of Gal(Q(√10)/Q)?

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Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

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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: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

Q: Let G be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?

A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…

Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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Q: c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

A: #Dear user there is a mistake in the question the assumption is for the element c and d of a group…

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: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

Q: (a) Draw the lattice of subgroups of Z/6Z. (b) Repeat the above for the group S3.

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Q: 5. Let H and K be normal subgroups of a group G such that H nK = {1}. Show that hk = kh for all h e…

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

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Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…

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Q: 24, Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then a and b are…

A: Given:  Let G be a group. ZG be its center.  We know that ZG=z∈G: ∀g∈G,zg=gz ....i First we will…

Q: (a) Compute the list of subgroups of the group Z/45Z and draw the lattice of subgroups. (prove that…

A: In the given question we have to write all the subgroup of the group ℤ45ℤ and also draw the the…

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: Consider the subgroup H = (i) of the the group (C\{0},.). LIT TT TT?

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Q: 4. Let (G, *) be a group of order 231 = 3 × 7 × 11 and H€ Syl₁₁(G), KE Syl, (G). Prove that (a). HG…

A: The Sylow theorems are a fixed of theorems named after the Norwegian mathematician Peter Ludwig…

Q: Let G be a group, and N ⊆ Z(G) be a subgroup of the center of G, Z(G). If G/N, the quotient group is…

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Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

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: 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: Let be a group and Ha normal subgroup of G. Show that if y.VEG such that xyEH then yx EH

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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: 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: a. If G is a group of order 175, show that GIH=Z, where H is a normal subgroup of G. b. Show that Z…

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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: 2. If H and K are subgroups of an abelian group G, then HK = {hk | h e H and k e K} is a subgroup of…

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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: 2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H…

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Q: The group (Z, t6) contains only 4 subgroups

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