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I have seen many questions of radical of ring, nil radical Jacobson radical etc on this site like-

$(1)$ Radical of a ring.

$(2)$ Radical of a quotient Lie algebra, etc, but do not got my answer. My problem is

I have studied two definition of Semi simple group:

$(1)$ A group $G$ is said to be semi simple if it can be written as direct product of simple groups.

$(2)$ A finite group $G$ is said to be semi simple if its Radical is trivial, where radical of a group is the largest solvable normal subgroup of the group $G$.

I have seen the second definition in the Algebra $1$ by Ramjilal (https://www.springer.com/gp/book/9789811042522). I wants to prove that these two definitions coincides, but do not know that how to approach. In short I want to prove that if a group $G$ has no normal solvable subgroup then it can be written as internal direct product of its simple normal subgroups. But I do not have any idea that how to do it?

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MANI
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  • Please someone have a look on my question. At least give me some references. – MANI Nov 20 '19 at 06:29
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    These two definitions aren't equivalent. For example, the only non-trivial normal subgroup normal subgroup of $S_5$ is $A_5$, but $S_5$ is not a direct product of simple groups. Based on this reference, it seems that there is no uniform definition of semisimplicity for a finite group. – Mathmo123 Nov 20 '19 at 07:23
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    Definition (2) is really awkward. It's made in analogy with Lie algebras in char. zero, but it doesn't work well. Better call such groups "radical-free". – YCor Nov 20 '19 at 10:31
  • @YCor You saying good, but its authors choice. I was studying the book (written above) and confused with this definition. – MANI Nov 20 '19 at 10:41
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    @MANI Definition (1) is not appropriate, as general (infinite) direct products are difficult to handle and describe. The correct definition is that $G$ is semisimple if it is isomorphic to a restricted direct sum of simple groups, where the construction of restricted direct sums more generally applies to the category of monoids and is an immediate generalisation of the notion of direct sum of modules (although unlike this latter notion the restricted direct sum no longer is a direct sum in the categorical sense). – ΑΘΩ Nov 11 '20 at 04:04

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Let $G$ be the direct product of two cyclic groups of order $3$. Then $G$ is semi-simple according to the first definition, but not according to the second one.

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