Let H be a subgroup of G, let a be a fixed element of G, and let K be the set of all ele- ments of the form aha¯', where h E H. That is, K = {x E G|x = aha¯' for some h E H}. Prove or disprove that K is a subgroup of G.

Q: SUCH THAT LET H BE A PROPER SUBGROUP OF G V x,y € G-H, xy EH. PROVE THAT HAG.

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Q: . Let H be a subgroup of R*, the group of nonzero real numbers un- der multiplication. If R* C H C…

A: H be a subgroup of R*, the group of nonzero real numbers under multiplication. R+⊆ H ⊆ R*. To prove:…

Q: Let G be the subgroup of GL3(Z₂) defined by the set 100 a 10 bc1 such that a, b, c Z₂. Show that G…

A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

Q: Suppose that o: G→G is a group homomorphism. Show that () p(e) = ¢(e') (ii) For every gE G, ($(g))-1…

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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: Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Prove that if…

A: Given that, Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G.

Q: Prove that H x {1} and {1} x K are normal subgroups of H x K, that these subgroups general H x K,…

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Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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Q: Let H be a subgroup of a group G, S {Hx:xe G). nen prove that there is a homomorphism of G onto A(S)…

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Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

Q: (a) Prove that if K is a subgroup of G and L is a subgroup of H, then K x L is a subgroup of G x H.

A: The detailed solution of (a) is as follows below:

Q: (2) of order 5 is in H. Let G be a group of order 100 that has a subgroup H of order 25. Prove that…

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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: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: Let H be a subgroup of a group G, S {Hx: x e G}. %3D Then prove that there is a homomorphism ofG…

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Q: 2) Let be H. K be and gooup Subgroups f Relate Gu such That Na(H)=Nq(K). H and 'K.

A: Let G be a group. Let H and K be a subgroups of G such that NG(H)=NG(K) We relate H and K. Let G be…

Q: Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

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Q: Let G be a group, and let X be a set. Let I be the intersection of all subgroups of G that contain…

A: Let G be a group and X be a set in G. Suppose I is the intersection of all subgroups of G that…

Q: . Let H be a subgroup of a group G. Prove that the set HZG) = {hz | h E H, z E Z(G)} is a subgroup…

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Q: Let H & K be tavo to "HA'G, then Arnormal suograup subgroups group G. 4 Prove that HA K AG

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Q: Let a e G. Prove that $(a") = ¢(a)" for all n e Z.

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Q: Suppose that 0:G G is a group homomorphism. Show that () o(e) = 0(e) (i) For every gEG, ($(g))¯1…

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Q: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, H n K =…

A: 1. It is given that H∩K=x∈G|x∈H and x∈K Let  x, y∈H∩K ⇒x,y∈H and x,y∈K⇒xy-1∈H and xy-1∈K⇒xy-1∈H∩K…

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: Let G be a group and H, KG normal subgroups of G. Prove HnK≤ G.

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Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = –S, |S| = 3 and (S) =…

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Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

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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: 3) Show that the subgroup of Dg is isomorphic to V4.

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Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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Q: let H be a normal subgroup of G and let a belong to G. if th element aH has order 3 in the group G/H…

A: H is normal subgroup of G. And a belongs to G.  O( aH) = 3 in G/H and O(aH) in G/H divides  O(a) in…

Q: Consider the group G = Dg × Z/6Z, and let N = Dg x {0}. Show that N 4 G and that G/N = Z/6Z. Now…

A: Given is a group G and a subgroup N of G. First we use the definition of Normal subgroup to show…

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: 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: If H is a subgroup of G, then the index of H in G, written as (G : H), is the number of left (or…

A: Coset of H in G: Let H is a subgroup of the group G Then for any g∈G the set gH=gh : h∈H is called…

Q: Let G  be a group and H a normal subgroup of G. Show that if x,y in G such that xy in H then yx in H

A: We are given that H is a subgroup of G. ⇒) Assume H is a normal subgroup of G. So,…

Q: If op is a homomorphism of group G onto G with kernel K and Ñ is a normal subgroup of G. N = {x E G|…

A: What is Group Homomorphism: If there exists a bijective map θ:G→G' for two given groups G and G',…

Q: Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K…

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Q: 2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove…

A: Given   ϕ:G -&gt; G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

Q: 7. Let G be a group, and let g E G. Define the centralizer, Z(g), of g in G to be the subset Z(g) =…

A: let G be a group, and let g∈G. Define the centralizer, Zg of g in G to bethe subset…

Q: Let G and H be groups. Prove that G* = {(a, e) : a E G} is a normal subgroup of G × H.

A: We atfirst show that G* is a subgroup of G×H . Then we show that G* is normal in G×H

Q: Let G be a group and let H be a subgroup of G. For each g E G, we define the subset gHg- of G by…

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Q: Let G be a group and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

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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: 40) Let G be a group, let N be a normal subgroup of G and let G = and only if x-1y-1xy E N. (The…

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Q: |Let G and H be groups, and let 0 : G > H be a homomorphism. The set {x E G| 0(x) |e} is called the…

A: From the given information:Let G and H be the groups

Q: Let H and K be subgroups of a group G and assume |G : H| < +co. Show that |K Kn H G H\

A: Let G be a group and let H and k be two subgroup of G.Assume (G: H) is finite.

Q: Let H be a subgroup of G such that x^2 ∈ H for all x ∈ G, then show that H is a normal subgroup of…

A: H = {x² : x ∈ G} And, H &lt; G

Q: Let H be a subgroup of G, define C(H) the centralizer of H.

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.