* Group says it killed senior officer near Cairo
* Police raid militant hideout near Alexandria
* PM says state in "fierce war against terror"
* State extends Brotherhood crackdown, sentences 19 (Updates with claim of responsibility for bomb near Cairo)
By Tom Perry
CAIRO, April 23 (Reuters) - A senior police officer was killed near Cairo on Wednesday in a bomb attack claimed by a militant group, while security forces stormed a hideout used by another Islamist organisation near Alexandria in a raid that left an officer and a militant dead.
Militant violence has spiralled since last July, when the army toppled elected head of state Mohamed Mursi and the authorities launched a fierce crackdown on his supporters in the Muslim Brotherhood and other Islamist sympathisers.
The attacks underline lingering instability in Egypt ahead of a presidential election in May that Abdel Fattah al-Sisi, the former army chief who deposed Mursi, is expected to win. The prime minister said the state was in "a fierce war" on terror.
The police officer killed near Cairo was named as Brigadier General Ahmed Zaki. State media said he was killed outside his home in 6th of October City, 32 km (20 miles) outside Cairo, when a bomb placed under his car went off.
Two conscript policemen were wounded in the bombing. A militant group called Ajnad Misr, or Soldiers of Egypt, said it carried out the attack in a statement posted on a Facebook account in its name that has carried past statements.
The post included a photo of a man said to be Zaki on his way to his vehicle, describing him as "the criminal brigadier general in the (security) force for killing protesters".
Mursi's removal from power last summer after mass protests against his rule tipped Egypt into the worst internal strife of its modern history. Hundreds of his supporters were killed by security forces as they broke up their protest camps.
Militant attacks since then have killed around 500 people, mostly policemen and soldiers. The threat has been compounded by a flow of weapons from neighbouring Libya.
The Interior Ministry said the hideout targeted by police at dawn on Wednesday near Alexandria was used by Ansar Bayt al-Maqdis, or Supporters of Jerusalem, the group behind some of the deadliest attacks of the last nine months.
The militants had opened fire on the security forces as they arrived at the hideout in Borg El Arab, some 45 km (28 miles) south-west of Alexandria.
The police officer killed in the raid was named as First Lieutenant Ahmed Saad and the dead militant as Hassan Abdel Aal, a 25-year old from the Nile Delta province of Dakahlia.
Two other militants were detained, the ministry spokesman, Hany Abdel Latif, said in a televised statement. Footage broadcast on state TV appeared to show the body of a militant on the ground.
The militants were "among the dangerous elements of the terrorist group Ansar Bayt al-Maqdis, which was planning to target police and military facilities and the security forces", the ministry said. The police seized weapons including explosive belts, automatic weapons, hand grenades and ammunition.
"A FIERCE WAR"
"The state ... is in a fierce war against the forces of terror and extremism that want to obstruct the country's path," Prime Minister Ibrahim Mahlab said in comments posed on an official Facebook page on Wednesday.
Ansar Bayt al-Maqdis has claimed responsibility for attacks including a failed attempt to blow up the interior minister last September, and large bomb attacks on police stations.
The United States this month designated Ansar Bayt al-Maqdis as a terrorist group. The group first emerged in the Sinai Peninsula in 2011 after the downfall of Hosni Mubarak. Since last summer, it has switched its focus from attacking Israel to targeting Egyptian security forces.
It has also claimed responsibility for a Feb. 16 attack that killed three South Korean tourists in the Sinai Peninsula.
The army-backed authorities have declared the Muslim Brotherhood a terrorist group and accuse it of a role in violence. The group says it is committed to peaceful activism.
The authorities are pressing a crackdown on the group that has led to the imprisonment of thousands of activists including most of its top leadership, who are standing trial in multiple court cases on charges that include treason.
A court in the Nile Delta city of Tanta sentenced 19 Mursi supporters on Wednesday to 10 years each in prison for crimes including membership of an outlawed group and violence.
In a separate case, a Cairo court sentenced 13 Brotherhood members to three years in prison each and fined each of them 100,000 Egyptian pounds ($14,300) for protesting without permission and attacking police last December, the state news agency reported. All were university students. (Additional reporting by Abdel Rahman Yousef in Alexandria and Ali Abdellati in Cairo; Writing by Tom Perry; Editing by Michael Georgy, Hugh Lawson and Sonya Hepinstall)