fix 4. Prove that H is a subgroup of G. Also prove (7) Let H≤ G, then the set aHa is a subgroup for each a € G, and HaHa-¹. TC

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Q: Let H be a subgroup of G. If a and b are elements of G such that aH = bH, then Ja| = |b|. %3D

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Q: 2. Describe the following sets: 47 η 6Z = Ζ: π ε 47 and πε 6Z)

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Q: 27. Let H be a subgroup of a group G and let g € G. Define a one-to-one map of H onto Hg. Prove that…

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Q: A={[₁]IXER,YER} | x R* А

A: A is not a subgroup of GL(2, ℝ) because it is not closed under matrix multiplication .

Q: 4. If H is a subgroup of G, then show that the set W = ngHg¹ is a normal 9€G subgroup of G.

A: Given That : H is a subgroup of G  To Show: The set W=∩g∈GgHg-1 is a normal Subgroup.

Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

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Q: d) G=(Q-(0}, ); H=Q+ e) G=(Zg, ); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under…

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Q: Let G = A4. (3) Find the subgroup H of order 4. (4) List all left cosets of H in G.

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Q: Find the number of G-orbits in X := {1,2,3, 4, 5, 6, 7, 8, 9, 10}, where G < S10 is the subgroup…

A: Solution:Let g1 = (1,2), g2 = (3,4,5) - and g3 = (6,7,8,9). All three cycles at sight are…

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Q: greup and Subgroup of G. Then the following Statements are equivalent- is a normal subgroup in Gj…

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Q: Let G = U56 and define two subgroups H₁ and H₂ of G by := H₁ := Calculate H₁ H₂|. {1, 5, 9, 13, 25,…

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Q: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

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Q: Let G be a group and a 2 G. Then the set = {a" : ne Z}is a subgroup of G. 48.

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Q: 2. Let G be a group. Let H be a subgroup of G, and let g € G. Show that the set -1 gH g-1 {ghg¹|he…

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A: We need to prove the above theorem.

Q: n. Is it like cosets or something，a clear

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Q: Question: let Gi and G2 be two greups the map defined byl find iabmarphism GixG

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Q: Consider set A that is shown below. Show that A is a subgroup of GL(2, ℝ) using the 2-step Subgroup…

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Q: need the answer as soon as possi

A: We will find all the subgroups of S3 and define a subgroup.