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A composition series for a group is a sequence of subgroups that connects the trivial group to the group itself. Each subgroup is normal in the next, and the quotient of successive terms is a simple group. Simple groups are those that cannot be broken down further, as they have no nontrivial, proper normal subgroups. For the group \( G \) of order 42, you can construct a composition series by first understanding its structure through prime factorization.

A composition series of \( G \) is:

• \( \{ e \} \triangleleft C_2 \triangleleft C_6 \triangleleft C_6 \times C_7 \)

Break it down with the following logic:

• Start with the trivial group \( \{ e \} \).
• \( C_2 \) is a subgroup of \( C_6 \), which is cyclic of prime order.
• \( C_6 \) is built from \( C_2 \times C_3 \).
• Combine \( C_6 \) with \( C_7 \) for \( C_6 \times C_7 \), that is isomorphic to the original group \( G \).

At each stage, quotients like \( C_2 \) and \( C_3 \) are simple because of their prime order, highlighting why this is a valid composition series.

Composition factors are the building blocks of the composition series. They are the quotients between successive terms in the series. For the Abelian group \( G \) of order 42, once the composition series is determined, you calculate these factors to simplify and comprehend the structure of the group.

From the series:

• \( \{ e \} \triangleleft C_2 \triangleleft C_6 \triangleleft C_6 \times C_7 \)

The composition factors are:

• \( C_2 \)
• \( C_3 \)
• \( C_7 \)

These factors are simple groups because their orders are prime numbers. Simple groups mean that they do not have any proper subgroups other than themselves and the trivial group. This simplicity aligns perfectly with the definition you strive to achieve, making composition factors a fundamental part of breaking down groups into understandable pieces.

Understanding the structure of the group \( G \) requires prime factorization. Prime factorization allows decomposition of a number into products of prime numbers, which is critical for understanding the layers within a group. With an order of 42, the prime factorization is:

\[ 42 = 2 \times 3 \times 7 \]

Why is this important?

• Each prime number corresponds to a cyclic group of that order.
• The group \( G \) is isomorphic to a direct product of such cyclic groups, as indicated by the Fundamental Theorem of Finite Abelian Groups.
• This theorem lets us see that \( G \) has the structure \( C_6 \times C_7 \), \( C_2 \times C_{21} \), or \( C_{42} \).

By breaking \( G \) down to these cyclic groups, you understand that each can further be divided according to its own prime factorization, revealing a potential composition series. Prime factorization thus serves as the foundation for defining and analyzing the group’s composition series and factors.