A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is a uniform spherical group and G is simple, prove that f is an amorphous H =< e >.

Q: (3) Write down all proper subgroups in the group Z3 × Z3.

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Q: subgroup of G. Then the following statements are

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Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

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: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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Q: Prove that every subgroup of the quaternion group Qs is normal. Deter mine all the quotient groups.

A: The given group is Q8=±1,±i,±j,±k. There are four non trivial subgroups namely: -1=1,-1.…

Q: -. Prove that H = {[17]: nez} is a cyclic subgroup of GL₂(R). IC I 61 10

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Q: (True/False) Suppose G is a group. Then : G x G →G, (x,y) → ay is a group homomorphism. (True/False)…

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Q: Q4: Let A and B be subgroups of a group to that A1 = 4 and 1B1-5- Show that AnB = {e}

A: Let A and B be the two subgroups of Group GSuch  thati.e., the Order of subgroup A is 4 and B is 5…

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

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Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: An element x of a group satisfies x2 = e precisely when x = x-1. Use this observation to show how…

A: So no. of the elements of order 2 has to be odd.Explanation:Step 1: a = at ein ide"&gt;