Is group of order 32 solvable if yes then prove with the help of normal series and define solvable group for that. 

Yes, a group of order 32 is **solvable**. To explain why, let's first go through the definitions and the concept of solvability.

### Definition: Solvable Group
A group \( G \) is called **solvable** if it has a subnormal series:
\[
\{e\} = G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \dots \triangleleft G_n = G
\]
where each \( G_i \) is a normal subgroup of \( G_{i+1} \), and the quotient groups \( G_{i+1}/G_i \) are **abelian**.

In simple terms, a group is solvable if you can build a series of normal subgroups such that the successive quotients are abelian.

### Proof That a Group of Order 32 is Solvable

By the **Burnside's theorem**, if a group has order \( p^a q^b \) (i.e., its order is the product of powers of primes), where \( p \) and \( q \) are primes, then the group is solvable. 

In our case, the order of the group is 32, which is \( 2^5 \). Since it's a prime power (i.e., a group whose order is a power of a single prime), we can use a key result from group theory:

- **All groups of prime power order are solvable**.

This is because such groups have a non-trivial center, and you can construct a normal series where the successive quotients are abelian.

#### Steps to Construct the Normal Series:

Let \( G \) be a group of order 32. By the class equation and the properties of groups of prime power order, we know that \( G \) has a non-trivial center \( Z(G) \). Here's how the proof proceeds:

1. **Step 1: Existence of a non-trivial center**
   Since \( G \) is a \( p \)-group (in this case, a 2-group), it has a non-trivial center \( Z(G) \). The center \( Z(G) \) is a normal subgroup of \( G \) because the center of any group is normal in the group.

2. **Step 2: Quotient group is of smaller order**
   The quotient group \( G/Z(G) \) has order less than 32 (specifically, order \( 32/|Z(G)| \)). Since \( Z(G) \) is non-trivial, \( G/Z(G) \) has order a power of 2 (specifically, less than 32). This process can be repeated for \( G/Z(G) \), since it's also a \( p \)-group.

3. **Step 3: Continue the process**
   We continue this process of finding normal subgroups and constructing quotients. Because the order of each successive quotient group decreases, eventually, we reach the trivial group \( \{e\} \).

4. **Step 4: Abelian quotients**
   At each stage, the quotient groups formed (such as \( G/Z(G) \)) are abelian because we are dealing with \( p \)-groups and the successive centers/quotients are abelian.

5. **Step 5: Finite process**
   This is a finite process, and hence, we can construct a series of normal subgroups where the quotients are abelian.

Therefore, by following this procedure, we can form a normal series for \( G \) with abelian successive quotients, showing that the group is solvable.

### Conclusion:
Thus, a group of order 32 is solvable because it can be broken down into a normal series with abelian quotients, and it satisfies the conditions of Burnside's theorem for solvability.

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Let me know if you'd like further clarification or have any questions. 

Here are some related questions you might consider exploring:
1. How do we find the center of a group?
2. What is the class equation and how does it relate to group centers?
3. Can a group of prime power order ever be non-abelian?
4. What is the importance of Burnside's theorem in the theory of finite groups?
5. How do solvable groups relate to the concept of radical groups?

**Tip:** When proving solvability, constructing a normal series and checking the nature of the quotient groups is crucial. Always remember to use basic properties like the existence of the center in \( p \)-groups.

Is group of order 32 solvable if yes then prove with the help of normal series and define solvable group for that.

Discover why groups of order 32 are solvable, supported by a detailed proof using normal series. Learn about solvable groups, Burnside's theorem, and the process of constructing a normal series with abelian quotients. Ideal for math majors studying group theory.

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