9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba-1 E H ba E H O b-1a E H O None of these

Q: Let G be a group and a be an element of this group then necessarily: * Jal<|G| O lal2|G| O lal=|G|

A: In the given question we have to chose the correct option from the given options of the given…

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

A:

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

A:

Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…

A:

Q: (c) Prove that the intersection of any three subgroups is a subgroup while the union of two…

A:

Q: O In a group (G,x), if IGl= 37, the number of Passible Subgroups in G are

A: Note: Since you haven't mentioned which question you would like to get answered. We are providing…

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba EH O None of these…

A:

Q: Suppose that H is a subgroup of Z under addition and that H contains 250 and 350. What are the…

A: Given H be a subgroup of (Z,+) containing250 and 350Note that,  GCD(250 , 350)=1⇒by property of GCD,…

Q: 17. Let (G, *) be a group, and let H, K≤ G, H ≤K. Prove that (a). K/H AG/H ammad A Castanl/Collage…

A:

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if* O ba e H O b-1a e H…

A: We will use definition of left coset

Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is

A: It s given that H and K are subgroups of G, H=18 and K=30. Since H, K are subgroups, H∩K≤H  and…

Q: 1. Determine all subgroups of the group (U13, ·)

A: The sub group of U13 is to be determined.

Q: prove That :- let H and K be subgroups of agroupG of the m is ormal a HK is subgroup of G →1f one

A: Subgroup of a group G

Q: Let H be a subgroup of a group G and a, b EG. Then b E aH if and only if O None of these O ab EH О…

A: The solution is :

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: Let G be a group and a be an element of this group : then necessarily O laisIGI lal2/G] O lal=IG]

A: Given ,  Let G be a group and a be an element of this group

Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

A:

Q: 9) Let H be a subgroup of a group G and a, be G. Then a E bH if and only it O ba-1 eH O ba eH O b-1a…

A:

Q: Let H be a subgroup of a group G and a, b € G. Then b E aH if and only it O None of these O ab e H O…

A:

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: 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 H be a subgroup of a group G and a, bEG. Then bE aH if and only if * O None of these O ab e H O…

A: here option (c) is true.

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: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.

A:

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

A:

Q: 13. Find groups that contain elements a and b such that |al = \b| = 2 and a. Jab| = 3, b. Jab| = 4,…

A:

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*…

A:

Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')

A:

Q: Q1/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*)

A:

Q: 9. Prove that H ne Z} is a cyclic subgroup of GL2(R). . Subgraup chésed in Pg 34

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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: Show that if aEG, where G is a group and |a| = n then : %3D a' = a' if and only if n divides. -j

A:

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: 4) Let G. be Graup and aE G La> ç Cala)? give Is Prove OY Counter example G. H, k Such (2) Let be…

A: Centralizer of 'a' in G- Let a be a fixed element in a group G. Then the centralizer of 'a' in G is…

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

A:

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

A:

Q: Suppose that X and Y are subgroups of G if |X|=28 and |Y|=42, then what is

A: "According to Bartleby Guideline, Handwritten solution are not provided" Given,            |x|=28…

Q: 9) Let H be a subgroup of a group G and a, bEG. Then a e bH if and only if* O ba e H O None of these…

A:

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * None of these b-1a e H…

A: Second option is correct.

Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…

A:

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

A:

Q: et G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Since H is a cyclic group. and K is cyclic group. H=a,K=b If H⊂K then H∩K=H where H is cyclic If K⊂H…

Q: 1,) and (G2,*) be two groups and →G2 be an isomorphism. Then *

A: given that G1,. and G2,*are two groups  and φ:G1→G2  be an isomorphism

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

A:

Q: i have included a picture of the question i need help understanding.thank you in advance. please…

A: Let H and K are two subgroups of the group G.To show

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: : Show that in a group G, if a? = e,Vx E G, then G is a commutative. %3D

A:

Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution

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

A: Need to find intersection of subgroup